UNIVERSITY of CALIFORNIA Los Angeles on Bi-Free Probability And

UNIVERSITY OF CALIFORNIA Los Angeles

On bi-free probability and free entropy.

A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics

Ian Lorne Charlesworth

2017 c Copyright by Ian Lorne Charlesworth 2017 ABSTRACT OF THE DISSERTATION

On bi-free probability and free entropy.

Ian Lorne Charlesworth Doctor of Philosophy in Mathematics University of California, Los Angeles, 2017 Professor Dimitri Y. Shlyakhtenko, Chair

Free probability is a non-commutative analogue of probability theory. Recently, Voiculescu has introduced bi-free probability, a theory which aims to study simultaneously “left” and “right” non-commutative random variables, such as those arising from the left and right regular representations of a countable group. We introduce combinatorial techniques to characterise bi-free independence, generalising results of Nica and Speicher from the free setting to the bi-free setting. In particular, we develop the lattice of bi-non-crossing partitions which is deeply tied to the action of bi-freely independent random variables on a free product space. We use these techniques to show that a conjecture of Mastnak and Nica holds, and bi-free independence is equivalent to the vanishing of mixed bi-free cumulants vanishing. Moreover, we extend the theory into the operator-valued setting, introducing operator-valued cumulants which correspond to bi-freeness with amalgamation in the same way. Finally, we investigate regularity problems in algebras of non-commuting random variables. Using operator theoretic techniques show that in an algebra generated by non-commutative random variables which admit a dual system, any self-adjoint element with spectral measure singular with respect to Lebesgue measure is a multiple of 1. We are also able to slightly improve on prior results in the literature and show that any non-constant self-adjoint polynomial evaluated at a set of non-commutative random variables which are free, algebraic, and have ﬁnite free entropy must produce a variable with ﬁnite free entropy.

ii The dissertation of Ian Lorne Charlesworth is approved.

Milos D. Ercegovac

Sorin Popa

Dimitri Y. Shlyakhtenko, Committee Chair

University of California, Los Angeles

2017

iii For Sue, Dave, Laurel, and Kate.

iv TABLE OF CONTENTS

1 Introduction...... 1

1.1 Preliminaries...... 1

1.1.1 Probability...... 2

1.1.2 Free probability...... 4

1.1.3 Free cumulants...... 7

1.1.4 The incidence algebra of non-crossing partitions...... 9

1.1.5 Bi-free probability...... 11

1.1.6 Free convolutions...... 14

1.1.7 Fock space...... 16

1.1.8 Operator-valued free probability...... 17

1.1.9 Free entropy and regularity...... 18

2 Bi-free independence...... 23

2.1 Some combinatorial structures...... 23

2.1.1 Bi-non-crossing partitions...... 23

2.1.2 Shaded LR-diagrams...... 24

2.2 Computing joint moments of bi-free variables...... 28

2.3 The incident algebra on bi-non-crossing partitions...... 32

2.3.1 Multiplicative functions...... 33

2.4 Unifying bi-free independence...... 35

2.4.1 Summation considerations...... 35

2.4.2 Bi-free independence is equivalent to combinatorial bi-free independence. 40

2.4.3 Voiculescu’s universal bi-free polynomials...... 41

v 2.5 An alternating moment condition for bi-free independence...... 43

2.5.1 The equivalence...... 45

2.5.2 Conditional bi-freeness...... 47

2.6 Bi-free multiplicative convolution...... 50

2.6.1 The Kreweras complement...... 51

2.6.2 Cumulants of products...... 52

2.6.3 Multiplicative convolution...... 53

2.7 An operator model for pairs of faces...... 54

2.7.1 Nica’s operator model...... 55

2.7.2 Skeletons corresponding to bi-non-crossing partitions...... 56

2.7.3 A construction...... 57

2.7.4 The operator model for pairs of faces...... 63

2.8 Additional cases of bi-free independence...... 68

2.8.1 Conjugation by a Haar pair of unitaries...... 68

2.8.2 Bi-free independence for bipartite pairs of faces...... 70

3 Amalgamated bi-free probability...... 72

3.1 Bi-free families with amalgamation...... 72

3.1.1 Concrete structures for bi-free probability with amalgamation. . . . . 72

3.1.2 Abstract structures for bi-free probability with amalgamation. . . . . 77

3.1.3 Bi-free families of pairs of B-faces...... 81

3.2 Operator-valued bi-multiplicative functions...... 82

3.2.1 Deﬁnition of bi-multiplicative functions...... 82

3.3 Bi-free operator-valued moment function is bi-multiplicative...... 86

3.3.1 Deﬁnition of the bi-free operator-valued moment function...... 86

vi 3.3.2 Veriﬁcation of Property (iii) from Deﬁnition 3.2.1 for E...... 88

3.3.3 Veriﬁcation of Property (iv) from Deﬁnition 3.2.1 for E...... 92

3.4 Operator-valued bi-free cumulants...... 99

3.4.1 Cumulants via convolution...... 99

3.4.2 Convolution preserves bi-multiplicativity...... 100

3.4.3 Bi-moment and bi-cumulant functions...... 101

3.4.4 Vanishing of operator-valued bi-free cumulants...... 104

3.5 Universal moment polynomials for bi-free families with amalgamation. . . . . 106

3.5.1 Bi-freeness with amalgamation through universal moment polynomials. 106

3.6 Additivity of operator-valued bi-free cumulants...... 112

3.6.1 Equivalence of bi-freeness with vanishing of mixed cumulants. . . . . 112

3.6.2 Moment and cumulant series...... 113

3.7 Amalgamated versions of results from Chapter 2...... 115

3.7.1 Amalgamated vaccine...... 115

3.7.2 Amalgamated multiplicative convolution...... 117

4 An investigation into regularity and free entropy...... 118

4.1 Regularity results...... 118

4.1.1 Useful results from the literature...... 118

4.2 Algebraicity and ﬁnite entropy...... 119

4.2.1 Properties of spectral measures...... 120

4.3 Bi-free unitary Brownian motion...... 124

4.3.1 Free Brownian motion...... 125

4.3.2 The free liberation derivation...... 127

4.3.3 A bi-free analogue to the liberation derivation...... 128

vii 4.3.4 Bi-free unitary Brownian motion...... 131

viii ACKNOWLEDGMENTS

I would like to thank my advisor, Dima Shlyakhtenko, for years of invaluable advice and support. Without his guidance and encouragement, I would never have attained the expertise and will necessary to complete this dissertation. I am in his debt for all the help he has given me, with respect to mathematics and to my future career. I could not have asked for a better advisor.

I am grateful to Ben Hayes, Brent Nelson, and Paul Skoufranis, in whose footsteps I have followed. They have served as great role models for me in graduate school, providing me with friendship, mentoring, and collaboration. I wish to extend thanks as well to the many people who have provided me variously with advice, opportunities, suggestions, insight, and camaraderie: Andreas Aaserud, Michael Anshelevich, Ken Dykema, Todd Kemp, Brian Leary, Dave Penneys, Jesse Peterson, Sorin Popa, and Dan Voiculescu.

Last but by no means least, I am thankful to the many people who have provided me love and support over the years which has helped me stay motivated and focused: my parents, Sue and Dave; my sisters Laurel and Kate; and Nadia and Rami, Sean and Becky, Fred and Steph, Brittany, Jason, Alec, and Danny.

The content of [9] is included in Chapter 2; this paper was joint work with Brent Nelson and Paul Skoufranis and arose from useful discussions we shared while at the University of California, Los Angeles. Thus some parts of Chapter 2 were published in the Cana- dian Journal of Mathematics, http://dx.doi.org/10.4153/CJM-2015-002-6. [8] has been distributed between Chapters 2 and 3, and arose from continuations of those same discussions. The ﬁnal publication is available at Springer via http://dx.doi.org/10.1007/ s00220-015-2326-8. Some of the arguments from [10] appear in Chapter 4: the main result of that paper was discovered by Dima Shlyakhtenko and has not been included; the included pieces were arguments I discovered with his advice which were also included in the paper. The full publication appears at http://dx.doi.org/10.1016/j.jfa.2016.05.001. Finally, the preprint [7] contains content from Chapters 2 and 4.

I have received support from the Natural Sciences and Engineering Research Council of ix Canada in the form of scholarships, and the National Science Foundation through grants made to my advisor. I would also like to thank the Hausdorﬀ Research Institute for Math- ematics at the Universit¨atBonn for providing an excellent research environment during the von Neumann Algebras trimester program in the summer of 2016.

x VITA

2007-2012 B.Math. Honours Pure Mathematics & Honours Computer Science, Uni- versity of Waterloo, Waterloo, Ontario, Canada.

2012-2016 Natural Sciences and Engineering Research Council of Canada Postgradu- ate Scholarship.

2016-2017 U.C.L.A. Dissertation Year Fellowship.

PUBLICATIONS

A. Nica, I. Charlesworth, and M. Panju. Analyzing Query Optimization Process: Portraits of Join Enumeration Algorithms. IEEE 28th International Conference on Data Engineering (2012).

I. Charlesworth, B. Nelson, and P. Skoufranis. On Two-faced Families of Non-commutative Random Variables, Canadian Journal of Mathematics 67 (2015), no. 6, 1290-1325.

I. Charlesworth, B. Nelson, and P. Skoufranis. Combinatorics of Bi-Freeness with Amalga- mation, Communications in Mathematical Physics 338 (2015), 801-847.

I. Charlesworth and D. Shlyakhtenko. Free Entropy Dimension and Regularity of Non- commutative Polynomials, Journal of Functional Analysis 271 (2016), no. 8, 2274-2292.

I. Charlesworth. An alternating moment condition for bi-freeness. arXiv preprint 1611.01262 (2016).

xi I. Charlesworth, K. Dykema, F. Sukochev, and D. Zanin. Joint spectral distributions and invariant subspaces for commuting operators in a ﬁnite von Neumann algebra. arXiv preprint 1703.05695 (2017).

xii CHAPTER 1

Introduction.

Free probability is a non-commutative probability theory, introduced by Voiculescu in the 1980’s. While probability theory concerns itself with studying random variables – measurable functions on a probability space – non-commutative probability attempts to describe phenomena exhibited in algebras of non-commutative random variables, which are ∗-algebras equipped with positive states. The original motivations for developing free probability related to the study of von Neumann algebras, but it has since been connected to other ﬁelds, such as random matrix theory and combinatorics. The theory of free probability has been developed through both analytic and combinatorial techniques, and the interplay between these two has has led to many beautiful results.

Conventions. We will adopt the following conventions and notation:

• We take inner products to be linear in the second entry and conjugate linear in the ﬁrst.

• Given n ∈ N, we will denote by [n] the n-element set {1, 2, . . . , n}.

• Unless we explicitly state otherwise, all algebras are unital ∗-algebras and all subalgebras include the unit of the larger algebra.

1.1 Preliminaries.

We will provide a brief review of several ideas essential to free probability, which will be of use to us later on. First, though, we will provide a lens through which probability theory

1 may be viewed, which clariﬁes several analogies to free probability.

1.1.1 Probability.

To begin, we will recall some basics of probability. Our presentation will diﬀer from the usual,

in order to make the generalisation to the free setting more natural. Suppose that (Ω, F, P) is a probability space (so Ω is a set, F is a σ-algebra on F, and P is a positive measure of total mass 1), and let A = L∞(Ω, P) be the algebra of essentially bounded functions on Ω (i.e., the algebra of bounded random variables on Ω). Then A can be embedded in the

bounded operators on the Hilbert space L2(Ω, P) via pointwise multiplication. Notice that the expectation function E : A → C can be extended to B(L2(Ω, P)) as the vector state corresponding to 1 ∈ L2(Ω, P): for any X ∈ A,

E[X] = h1,X(1)i .

We will often forget the underlying probability space (Ω, P) and consider normed commuta-

tive unital ∗-algebras equipped with states (i.e., positive linear functionals which map 1A to 1); overloading terminology, we will also refer to such a pair as a probability space.

(ι) Recall that subalgebras (A )ι∈I are independent if for any X1,...,Xn ∈ A with Xk ∈

(ik) A such that i1, . . . , in are distinct, we have the identity

E[X1 ··· Xn] = E[X1] ··· E[Xn].

˚ ˚ ˚ Notice that it is equivalent to ask that this identity holds only for X1,..., Xn when E[Xk] = 0 ˚ for each k. Indeed, given arbitrary X1,...,Xn, we can set Xj = Xj − E[Xj] and obtain the relation by expanding the multiplication in

h ˚ ˚ i E X1 ··· Xn = 0.

Random variables are then said to be independent if they generate independent subalgebras

(ι) of A. Note that the value of E on the algebra generated by the (A )ι∈I is completely prescribed by its values on the individual A(ι) and the condition that they be independent. In fact, our condition for independence still makes sense and still prescribes joint moments 2 in terms of pure ones if we take A(ι) ⊂ A which commute with one another, but may not themselves be commutative!

Algebras of random variables may always be embedded into a larger algebra in a manner which makes them independent. The usual approach is to take products on the level of probability spaces (Ω, F, P); however, since we have forgotten the underlying probability space, this is not possible in our context. We will perform the construction in a more complicated manner, which will be easily duplicated in the non-commutative case.

A vector space with specified state vector is a triple (V, V˚ , ξ) where V is a vector space ˚ such that V = V ⊕ Cξ. We define a state ϕξ on L(V ), the space of linear operators on V , by ˚ taking ϕξ(T ) to be the unique element of C so that T (ξ) ∈ V +ϕξ(T )ξ. This makes (L(V ), ϕξ) into a probability space (and any probability space (A, E) can be realized in this form by ˚ letting V = A, ξ = 1A, and V = ker E, then having A act on V by multiplication). Now suppose that for each ι ∈ I we have A(ι) ⊂ L(V (ι)) with state vectors ξ(ι); for simplicity, assume I = {1, . . . , n} is finite. Consider the space V = V (1) ⊗ · · · ⊗ V (n) and let ξ = (1) (n) ˚ ˚(j) N (ι) ξ ⊗ · · · ⊗ ξ , with V = spanj∈I V ⊗ ι6=j V ; note that ϕξ = ϕξ(1) ⊗ · · · ⊗ ϕξ(n) . Then A(ι) embeds in L(V ) as 1 ⊗ · · · ⊗ 1 ⊗ A(ι) ⊗ 1 ⊗ · · · ⊗ 1 in a state preserving way, and

these embeddings are independent from one another under ϕξ.

(ι) This gives us another way of recognising independence: subalgebras (A )ι∈I of (A, ϕ) are independent if there exist vector spaces with speciﬁed state vectors (V (ι), V˚(ι), ξ(ι)) and embeddings ρ(ι) : A(ι) → L(V (ι)) such that the following diagram commutes:

O (ι) ev A A ι∈I

N (ι) ι∈I ρ ϕ ! ϕ O (ι) ξ L V C ι∈I

3 1.1.2 Free probability.

We will be working in the context of a non-commutative algebra, so it will be useful to have some operator algebraic concepts. A (concrete unital) C∗-algebra is a ∗-subalgebra of B(H) containing the identity operator which is closed under the operator norm topology. A (concrete) von Neumann algebra is a ∗-subalgebra of B(H) which is closed under the weak operator topology, or equivalently is closed under the strong operator topology, or equivalently is equal to its own bicommutant: that is, if A0 ⊂ B(H) is deﬁned as {x ∈ B(H): xa = ax ∀a ∈ A}, a von Neumann algebra satisﬁes A00 = A. A von Neumann

algebra is called a factor if its centre is C1. A state on a ∗-algebra A is a linear map ϕ : A → C such that ϕ(1) = 1 and ϕ is positive in the sense that ϕ(a∗a) ≥ 0 for a ∈ A.A state is

• faithful if ϕ(a ∗ a) = 0 implies a = 0;

• normal if for every bounded increasing net aα → a, ϕ(a) = limα ϕ(aα); and

• tracial if ϕ(ab) = ϕ(ba).

When we deal with von Neumann algebras, we will usually assume that they are ﬁnite (in the sense that no projection is equivalent to a proper subprojection) and equipped with faithful normal tracial states. A ﬁnite factor which is not isomorphic to B(H) and admits a faithful

normal tracial state is said to be of type II1.

A non-commutative probability space is a pair (A, ϕ) where A is a unital ∗-algebra and

ϕ : A → C is a state. Elements of A can be thought of as non-commutative random variables. Note that we make no assumption that A is non-commutative, so L∞(Ω, P) is a non-commutative probability space; the “non-commutative” merely references the potential of A to contain elements which do not commute, in much the same way that “densely-deﬁned unbounded operators” may be deﬁned everywhere and bounded.

Suppose that A is a C∗-algebra, and x ∈ A is normal. Then by Gelfand duality, the C∗-algebra generated by x is isometrically isomorphic to the continuous functions on the

4 (compact) spectrum of x, which when coupled with the spectral measure of x becomes a probability space. Thus normal elements of C∗ non-commutative probability spaces are

random variables in a very concrete sense. Given a collection of elements x1, . . . , xn ∈ A, their law is the map

µ(x1,...,xn) : C hX1,...,Xni → C

given by µ(x1,...,xn)(P (X1,...,Xn)) = ϕ(P (x1, . . . , xn)). When x1, . . . , xn are commuting normal elements of a C∗ algebra, this corresponds to their joint law in the usual probability sense.

We are now ready to introduce free independence as a non-commutative analogue of independence, using the final picture of independence as our starting point. For this we introduce the free product of vector spaces with specified state vectors, an analogue to the tensor product in much the same way that the free product of groups is a non-commutative (ι) ˚(ι) (ι) analogue of their direct product. Given a collection (V , V , ξ )ι∈I of vector spaces with specified state vectors (so once again V (ι) = V˚(ι) ⊕ Cξ(ι)) we define their free product to be the triple (V, V˚ , ξ) where V = V˚ ⊕ Cξ and M M V˚ := V˚(i1) ⊗ · · · ⊗ V˚(in).

n≥1 i16=i26=···6=in

Here the condition i1 6= i2 6= · · · 6= in requires only that adjacent indices be unequal. We denote ∗ V (ι) := V and ∗ ξ(ι) := ξ. ι∈I ι∈I (ι) (i) Suppose V = ∗ι∈I V ; then we can embed L(V ) in L (V ) as follows. First, we deﬁne

M M (i ) (i ) V (`, j) := Cξ ⊕ V˚ 1 ⊗ · · · ⊗ V˚ n . n≥1 j6=i16=···6=in The space V (`, j) can be thought of as words in V which do not begin with a letter from V˚(j). (j) ∼ Then we have V ⊗ V (`, j) = V via the map Wj which makes the obvious identiﬁcations:

(j) Wj(ξ ⊗ ξ) = ξ

˚(i1) ˚(in) ˚(i1) ˚(in) Wj(ξ ⊗ (V ⊗ · · · ⊗ V )) = V ⊗ · · · ⊗ V ˚(j) ˚(j) Wj(V ⊗ ξ) = V

˚(j) ˚(i1) ˚(in) ˚(j) ˚(i1) ˚(in) Wj(V ⊗ (V ⊗ · · · ⊗ V )) = V ⊗ V ⊗ · · · ⊗ V . 5 (ι) (j) (ι) −1 This gives us a left representation λ : L(V ) → L(V ) via λ (T ) = Wj ◦ (T ⊗ 1) ◦ Wj . Note that we may perform a similar decomposition by factoring V (j) out of V on the right,

(j) say Uj : V (r, j) ⊗ V → V (where V (r, j) is deﬁned analogously to be words which do not end with a letter from V (j)), and produce a right representation on the free product via

(ι) −1 ρ (T ) = Uj ◦ (1 ⊗ T ) ◦ Uj .

(ι) Now, given ∗-subalgebras (A )ι∈I of a non-commutative probability space (A, ϕ), we say they are freely independent (or free) if there exist vector spaces with speciﬁed state vectors (V (ι), V˚(ι), ξ(ι)) and embeddings π(ι) : A(ι) → L(V (ι)) such that the following diagram commutes: ∗ A(ι) ev ι∈I A

(ι) (ι) ∗ι∈I λ ◦ π ϕ

ϕξ L ∗ V (ι) ι∈I C

(ι) Here ∗ι∈I A we mean the formal algebraic free product consisting of reduced words with

(ι) (ι) (ι) letters coming from the A , and by ∗ι∈I λ ◦π we mean the unique algebra homomorphism

(ι) (ι) (ι) (ι) which is λ ◦ π when restricted to A . Notice that given a family of algebras (A )ι∈I we can use this construction to produce a state on their algebraic free product with respect to which they are free.

Example 1.1.1. Suppose that Γ1,..., Γn are countable groups and Γ = Γ1 ∗ · · · ∗ Γn. Let

τ : C[Γ] → C be the linear functional given by τ(e) = 1 and τ(g) = 0 for other g ∈ Γ. Then n (C[Γ], τ) is a non-commutative probability space, and the family of algebras (C[Γi])i=1 are freely independent. Notice that C[Γ] can be represented on `2(Γ) via left translation, and

τ(x) = hδ0, x · δ0i.

Unfortunately, this is not a very practical condition for checking free independence. For- tunately, it can be reformulated in terms of conditions on mixed moments which are far more approachable.

(ι) Proposition 1.1.2. Suppose (A )ι∈I is a family of unital ∗-subalgebras within a non- commutative probability space (A, ϕ). Then the family is freely independent if and only 6 (ij ) if whenever a1, . . . , an ∈ A are such that aj ∈ A with i1 6= i2 6= · · · 6= in and ϕ(aj) = 0

for each j, we have ϕ(a1 ··· an) = 0. That is, if and only if alternating products of centred variables are centred.

(ι) (ι) Proof. Suppose (A )ι∈I are free, with their freeness realised via ∗ι∈I V , and let a1, . . . , an

(in) (in) ˚(in) be as in the statement of the proposition. Notice that vn := λ ◦ π (an)ξ ∈ V as

(in−1) (in−1) ˚(in−1) ˚(in) ϕ(an) = 0. Similarly, vn−1 := λ ◦ π (an−1)vn ∈ V ⊗ V since ϕ(an−1) = 0.

˚(i1) ˚(in) Continuing this in the obvious manner, we ﬁnd v1 ∈ V ⊗ · · · ⊗ V and hence (ι) (ι) ϕξ ∗ λ ◦ π (a1 ··· an) = 0. ι∈I

Now, on the other hand, suppose that all alternating products of centred variables are themselves centred. Notice that this determines all mixed moments in terms of pure moments

(ij ) from the families: given arbitrary b1, . . . , bn with bj ∈ A and i1 6= i2 6= · · · 6= in, we ﬁnd

ϕ ((b1 − ϕ(b1)) ··· (bn − ϕ(bn))) = 0;

expanding this gives us an equation for ϕ(b1 ··· bn) in terms of products of mixed moments with fewer terms, which may themselves be recursively reduced to pure moments. Thus

(ι) there is a unique state on ∗ι∈I A which satisﬁes this condition. However, if we let µ be the

(ι) state on ∗ι∈I A which makes the individual algebras free, we find µ satisfies the alternating moment condition by the first part of this argument. Then by uniqueness, µ is equal to ϕ

(ι) and so the (A )ι∈I are free with respect to ϕ.

1.1.3 Free cumulants.

The free cumulants were introduced by Speicher to better understand the combinatorics of free independence, and free additive convolution [20,21].

Deﬁnition 1.1.3. A partition of a ﬁnite set S is a set π = {B1,...,Bn} of non-empty Sn subsets of S which are pairwise disjoint with S = i=1 Bi. The sets Bi are referred to as the

blocks of π. For x, y ∈ S we write x ∼π y if there is some B ∈ π with x ∈ B, y ∈ B. We deﬁne P(n) := {π : π is a partition of [n]}. 7 A partition π ∈ P(n) is said to be non-crossing if whenever 1 ≤ i < j < k < ` ≤ n are such that i ∼π k and j ∼π `, we have j ∼π k. The set of non-crossing partitions of n elements is denoted NC(n) := {π ∈ P(n): π is non-crossing}.

Non-crossing partitions are so named because they are exactly those which can be diagrammatically represented above the number line without needing to draw crossing blocks.

Example 1.1.4. There are 15 elements in P(4), of which 14 are non-crossing.

1 2 3 4 1 2 3 4 1 2 3 4 The partitions {{1, 4} , {2, 3}} and {{1, 2, 4} , {3}} are non-crossing, while {{1, 3} , {2, 4}} is the only element of P(4) which is not non-crossing.

The set P(n) is partially ordered by reﬁnement: given π, σ ∈ P(n) we say σ reﬁnes π, and write σ ≺ π, if every block of π is a union of blocks of σ. This ordering makes P(n) into a lattice with minimum element {{1} ,..., {n}} and maximum element {[n]}, denoted respectively 0P(n) and 1P(n) (although when context makes it clear we may instead write 0n and 1n).

We are now ready to deﬁne the free cumulants.

Deﬁnition 1.1.5. Suppose (A, ϕ) is a non-commutative probability space. The free cumu-

n lants are a family of linear functionals (κn)n≥1 with κn : A → C deﬁned recursively by the moment-cumulant formula: for any a1, . . . , an ∈ A, X Y ϕ(a1 ··· an) = κk (ai1 , . . . , aik ) .

π∈N C(n) {i1<···

We may denote the product appearing in this relation as κπ(a1, . . . , an) when it is convenient to do so, and given B = {i1 < ··· < ik} ⊂ [n] we may abbreviate κk(ai1 , . . . , aik ) =:

κk(aB). Likewise, it will often be convenient to write Y ϕ(aB) := ϕ(ai1 ··· aik ) and ϕπ(a1, . . . , an) := ϕ(aB). B∈π 8 (ι) Theorem 1.1.6 ([20]). Let (A )ι∈I be subalgebras of a non-commutative probability space

(ι) (A, ϕ). Then (A )ι∈I are free if and only if all mixed cumulants vanish, i.e., whenever

(ij ) a1, . . . , an ∈ A with aj ∈ A and ij 6= ik for some j, k, we have

κn(a1, . . . , an) = 0.

1.1.4 The incidence algebra of non-crossing partitions.

To better understand the cumulant functionals, Speicher in [20] made a study of the incidence algebra of non-crossing partitions. We provide an overview of some of those arguments, as they will provide a starting point for some of our arguments later on.

Proposition 1.1.7 ([20, Proposition 1]). Suppose σ ≺ π ∈ N C(n). Then there is a canonical decomposition of [σ, π] := {ρ ∈ N C(n): σ ρ π} as a product of full lattices:

k Y NC(aj). j=1

Sketch of proof. First, we notice that we may decompose the intervals into pieces corresponding to the blocks of π: Y Y [σ, π] ∼= [{C ∈ σ : C ⊂ B} , {B}] ⊂ NC(#B). B∈π B∈π (Here we have abused notation slightly: {B} ∈/ N C(#B) in general since the labels are wrong; this can be ﬁxed in the obvious way, by relabeling with the numbers 1,..., #B while maintaining ordering.)

Now, given an interval [σ, 1n], if possible we select a block {i1 < ··· < ik} ∈ σ and

1 ≤ j < k so that ij + 1 < ij+1. We then take

σ0 = {C ∈ σ : C ⊂ [n] \{ij + 1, . . . , ij+1 − 1}} ,

σ1 = {C ∈ σ : C ⊂ {ij + 1, . . . , ij+1 − 1}} .

Note that the non-crossing condition ensures that every block of σ is accounted for above. We now make the identiﬁcation

∼ [σ, 1n] = [σ0, 1n−(ij+1−ij −1)] × [σ1 ∪ {{0}} , 1ij+1−ij ]. 9 We have added a dummy node to the right hand product to represent the block which is

dividing the partition. The left hand interval here shoud be ignored if σ0 = 1n−(ij+1−ij −1).

This may be continued until we are left with a product of terms of the form [σ, 1n], with ∼ σ consisting of intervals. But in this case, [σ, 1n] = NC(#σ).

There is some potential ambiguity due to the fact that as lattices, [σ, π] ∼= NC(1)×[σ, π]; we resolve this by insisting that we only take copies of NC(1) when the last step in the decomposition requires us to do so.

Example 1.1.8.         ∼      , 17 =  , 14 ×  , 17 1 2 3 4 5 6 7 1 5 6 7 0 2 3 4

∼= NC(2) × N C(3)

The incidence algebra on the lattice of non-crossing partitions consists of the following set of functions: ( )

a IA(NC) := f : NC(n) × N C(n) → C f(σ, π) = 0 unless σ π . n≥1

We equip IA(NC) with the convolution product given as follows:

X (f ? g)(σ, π) := f(σ, ρ)g(ρ, π). σ≤ρ≤π

Deﬁnition 1.1.9. A function f ∈ IA(NC) is said to be multiplicative if whenever [σ, π] ∼= Qk j=1 NC(aj) as in Proposition 1.1.7, one has

k Y f(σ, π) = f(0aj , 1aj ). j=1

Notice that for a given sequence of numbers (xn), there is a unique multiplicative function

so that f(0n, 1n) = xn for every n.

Proposition 1.1.10 ([20, Proposition 2]). The convolution of two multiplicative functions is multiplicative. 10 ∼ Qk Sketch of proof. Suppose [σ, π] = j=1 NC(aj). Then

X (f ? g)(σ, π) = f(σ, ρ)g(ρ, π) σ≤ρ≤π

k   Y X =  f(0aj , ρ)g(ρ, 1aj )

j=1 ρ∈N C(aj ) k Y = (f ? g)(0aj , 1aj ) j=1

We next label some particular elements of IA(NC):    1 if σ = π  1 if σ π δNC(σ, π) := , and ζNC(σ, π) := .  0 otherwise  0 otherwise

One can check that ζNC is invertible and compute its inverse recursively; we deﬁne µNC, the

M¨obiusfunction, to be the inverse of ζNC under convolution, so µNC ?ζNC = δNC = ζNC ?µNC.

We can now relate this to cumulants in the following natural way: suppose a1, . . . , an ∈ A; then the moment-cumulant formula may be “convolved” with ζNC to give the following equivalent formulations:

X κn(a1, . . . , an) = ϕπ(a1, . . . , an)µNC(π, 1n), π∈N C(n) X κπ(a1, . . . , an) = ϕσ(a1, . . . , an)µNC(σ, π). σ∈N C(n) σ≤π

Moreover, given a ∈ A, if m and k are the multiplicative functions corresponding to the

n n sequences (ϕ(a ))n and (κn(a, . . . , a))n in the sense that m(0n, 1n) = ϕ(a ) and k(0n, 1n) =

κn(a, . . . , a), we ﬁnd that m = k ? ζNC and k = m ? µNC.

1.1.5 Bi-free probability.

In Subsection 1.1.2, we remarked that given a collection of algebras represented on vector spaces with speciﬁed state vectors, we could as easily represent them on the right of the 11 free product of those vector spaces as on the left. Pondering this situation, Voiculescu introduced in [30] the concept of bi-free independence as an independence relation on pairs of sub-algebras in a non-commutative probability space. Bi-free probability aims to study simultaneously the behaviour of “left” and “right” variables.

Deﬁnition 1.1.11. Suppose (A, ϕ) is a non-commutative probability space. A pair of

faces in A is a pair (A`, Ar) of unital ∗-algebras together with a pair of unital embeddings

α` : A` → A, αr : Ar → A, though we will often take A`, Ar ⊂ A with the inclusion map for simplicity.

(ι) (ι) A family of pairs of faces (A` , Ar ) is said to be bi-free if there exist vector spaces ι∈I (ι) ˚(ι) (ι) (ι) (ι) (ι) (ι) with speciﬁed state vectors (V , V , ξ ) and embeddings π` : A` → L(V ) and πr : (ι) (ι) Ar → L(V ) such that the following diagram commutes:

∗ (α(ι) ∗ α(ι)) (ι) (ι) ι∈I ` r ∗ A` ∗ Ar A ι∈I

(ι) (ι) (ι) (ι) ∗ι∈I (λ ◦ π` ) ∗(ρ ◦ πr ) ϕ

ϕξ L ∗ V (ι) ι∈I C

(ι) (ι) Notice that once again, given a family of pairs of faces (A` , Ar ) we can use this ι∈I construction to produce a state on their algebraic free product with respect to which they

(ι) (ι) (ι) (ι) are bi-free. Given maps ϕ : A` ∗ Ar → C we will denote this bi-free state by ∗∗ι∈I ϕ .

Example 1.1.12. As in Example 1.1.1, let Γ1,..., Γn be countable groups, and

Γ = Γ1 ∗ · · · ∗ Γn.

2 Let α`, αr : C[Γ] → B(` (Γ)) be the left and right regular representations of Γ, respectively, (i) (i) 2 and set α` = α`|C[Γi], αr = αr|C[Γi]. Equip B(` (Γ)) with the state τ(x) = hδ0, x · δ0i. Then n (i) (i) 2 the family of pairs of faces α` (C[Γi]), α` (C[Γi]) is bi-free in B(` (Γ)). i=1

Proposition 2.9 of [30] demonstrates that checking for bi-freeness in a single free product (ι) (ι) representation suﬃces, and shows that there is a unique joint law on (A` , Ar ) which ι∈I 12 makes them bi-free. In particular, it suﬃces to check bi-free independence in the following (ι) (ι) universal representation: suppose (A` , Ar ) are a family of pairs of faces in a non- ι∈I (ι) (ι) (ι) commutative probability space (A, ϕ). Take V = A` ∗ Ar , a free product of algebras viewed as a vector space, and set ξ(ι) = 1 ∈ V (ι), V˚(ι) = ker ϕ(ι), where ϕ(ι) is the restriction

(ι) (ι) (ι) of ϕ to the algebra generated by A` and Ar , viewed as a map on V by evaluation in A. (ι) (ι) (ι) (ι) (ι) Then let π` , πr be the left actions of A` and Ar on V . (We do want the left action (ι) (ι) of Ar here; the fact that it is a right face will be reﬂected in its action on ∗ι∈I V .) Let (i) (i) (i) (i) (i) (i) qi = λ if χ(i) = `, and qi = ρ if χ(i) = r, where λ , ρ : L(V ) → L ∗ι∈I V are as (ι) (ι) (ι(i)) above. Then (A` , Ar ) are bi-free if and only if for every z1, . . . , zn with zi ∈ Aχ(i) , ι∈I we have (ι) (ι(1)) (ι(n)) ϕ(z1 ··· zn) = ∗ ϕ qi ◦ π (z1) ··· qi ◦ π (zn)1 . ι∈I χ(1) χ(n)

Further, some relations between bi-free independence and both free and classical indepen- (ι) (ι) dence were established in [30]. In particular, if (A` , Ar ) are bi-freely independent, ι∈I (ι) (ι) we have that (A` )ι∈I are free, (Ar )ι∈I are free, and if I,J ⊂ I are disjoint, the algebras (ι) (ι) generated by (A` )ι∈I and (Ar )ι∈J are classically independent. Moreover, if all the right (ι) (ι) faces are C then bi-freeness of (A` , Ar ) is equivalent to freeness of the left faces; if all ι∈I (ι) (ι) the left faces are C then bi-freeness of (A` , Ar ) is equivalent to freeness of the right ι∈I (1) (2) (1) faces; and (A` , C) is bi-free from (C, Ar ) if and only if A` is classically independent from (2) Ar .

In [14], Mastnak and Nica introduced a set of linear functionals which they conjectured to play the role of bi-free cumulants, which we will now introduce. Their cumulants are indexed not just by a number of arguments, but also by a choice of left or right for each, to account for the extra dynamics present in the bi-free setting.

Let n ∈ N, and take χ :[n] → {`, r}. We then deﬁne the permutation sχ as follows: if −1 −1 χ (`) = {i1 < ··· < ik} and χ (r) = {ik+1 > ··· > in}, then sχ(j) := ij.

Deﬁnition 1.1.13. Let (A, ϕ) be a non-commutative probability space. The (`, r)-cumulant functionals (or bi-free cumulants, or when context makes it clear, cumulants) are the maps

n (κχ : A → C)n≥1,χ:[n]→{`,r} , 13 which are deﬁned recursively by the moment-cumulant relation:

X Y ϕ(z1 ··· zn) = κχ|B (zB). π∈P(n) B∈π −1 sχ ·π∈N C(n)

As in the free case, we may denote the product on the right hand side by κπ(z1, . . . , zn).

Taking inspiration from Theorem 1.1.6, Mastnak and Nica deﬁned a family of pairs of (ι) (ι) faces (A` , Ar ) in a non-commutative probability space (A, ϕ) to be combinatorially ι∈I bi-free if whenever z1, . . . , zn ∈ A, i1, . . . , in ∈ I are not all equal, and χ :[n] → {`, r} with

(ij ) zj ∈ Aχ(j), it follows that

κχ(z1, . . . , zn) = 0.

Moreover, they conjectured that combinatorial bi-free independence was equivalent to bi-free independence.

Notice that the bi-free cumulants corresponding to χ ≡ ` are just the free cumulants.

1.1.6 Free convolutions.

Free probability provides us with operations deﬁned on the set of laws of random variables. Indeed, suppose that a, b ∈ A are free in some non-commutative probability space. We then

deﬁne the additive convolution of their laws to be the law µa µb := µa+b : C hXi → C; since all moments of a+b may be computed from the laws of a and b, this is well-deﬁned and doesn’t depend on the realization of a and b. Moreover, if a and b are self-adjoint elements of a C∗-algebra, then since a + b remains self-adjoint, free additive convolution gives a map from the space of pairs of laws of real-valued bounded random variables to the space of laws of real-valued bounded random variables.

The situation of multiplicative convolution is slightly more intricate. Once again, given a, b ∈ A which are free, we deﬁne the multiplicative convolution of their laws to be the law

µa µb := µab. Since the joint law of a and b is tracial (which can be checked using the cyclic symmetry of the non-crossing partitions) we ﬁnd that is commutative. If a and b ∗ are unitaries in a C -algebra, then so is ab, and we ﬁnd maps pairs of distributions on T 14 to distributions on T. Similarly, if a and b are positive, then a1/2ba1/2 is positive and has the same distribution as ab (though not the same ∗-distribution, as the former is self-adjoint while the latter is not); thus takes pairs of bounded distributions on R+ to bounded distributions on R+. These operations can be extended to the setting of distributions without compact support, but we do not need these technicalities here; see, e.g., [12].

The eﬀect of additive convolution may be described by the free cumulants. Indeed, if a, b are free, using the multi-linearity of cumulants and the vanishing of mixed cumulants, we ﬁnd that

κk(a + b, . . . , a + b) = κk(a, . . . , a) + κk(b, . . . , b).

In order to understand the combinatorics of the multiplicative convolution, we need to introduce the Kreweras complement; this description is due to Nica and Speicher in [17].

The Kreweras complement KNC : NC(n) → N C(n) is a lattice anti-isomorphism deﬁned ¯ as follows. Take π ∈ NC(n). Then KNC(π) is the maximum partition of {1¯,..., n¯} such ¯ that π ∪ KNC(π) is non-crossing as a partition of the ordered set {1 < 1¯ < 2 < ··· < n < n¯}. ¯ Finally, we deﬁne KNC(π) := B ⊂ [n]: {¯j : j ∈ B} ∈ KNC(π) ; that is, KNC(π) is the ¯ 2 non-crossing partition obtained by removing the bars from KNC(π). Note that KNC(π) corresponds to preforming a cyclic permutation on the labels of π.

Example 1.1.14. Suppose π = {{1} , {2, 4, 8} , {3} , {5} , {6, 7}}.

1 1¯ 2 2¯ 3 3¯ 4 4¯ 5 5¯ 6 6¯ 7 7¯ 8 8¯

Then we see that KNC(π) = {{1, 8} , {2, 3} , {4, 5, 7} , {6}}. On the other hand, we ﬁnd that

2 KNC(π) = {{8} , {1, 3, 7} , {2} , {4} , {5, 6}}.

The Kreweras complement is useful to us because it describes multiplicative convolution. Indeed, if a and b are free, we have

X κn(ab, . . . , ab) = κπ(a, . . . , a)κKNC(π)(b, . . . , b). π∈N C(n) 15 1.1.7 Fock space.

We mention here the notion of a Fock space, which will be useful for many future arguments and examples. Given a Hilbert space H, we deﬁne the Fock space F(H) as follows:

M ⊗n F(H) := CΩ ⊕ H . n≥1

The vector Ω ∈ F(H) is referred to as the vacuum vector. We refer to the vector state corresponding to the vacuum vector, ω : T 7→ hΩ,T Ωi, as the vacuum state.

Given ξ ∈ H, we deﬁne its left creation operator:

`(ξ)Ω = ξ and `(ξ)ξ2 ⊗ · · · ξn = ξ ⊗ ξ2 ⊗ · · · ξn.

The left annihilation operator corresponding to ξ is `∗(ξ), deﬁned as the adjoint of `(ξ), and satisﬁes

∗ ∗ ` (ξ)Ω = 0 and ` (ξ)ξ1 ⊗ · · · ⊗ ξn = hξ, ξ1i ξ2 ⊗ · · · ⊗ ξn,

where we interpret an empty product as the vector Ω. We also deﬁne analogously the right creation and annihilation operators r(ξ) and r∗(ξ).

To simplify notation, we will allow ourselves to factor tensor products over sums, and assume that tensoring with Ω corresponds to the identity map. Thus, for example,

(ξ1 + ξ2 ⊗ ξ3) ⊗ ξ4 = ξ1 ⊗ ξ4 + ξ2 ⊗ ξ3 ⊗ ξ4, and ξ ⊗ Ω = ξ = Ω ⊗ ξ.

We state the following useful proposition without proof:

Proposition 1.1.15. Suppose that (Hι)ι∈I are pairwise orthogonal subspaces of H. Then the

∗ ∗ pairs of faces (C h`(ξ), ` (ξ): ξ ∈ Hιi , C hr(ξ), r (ξ): ξ ∈ Hιi)ι∈I are bi-freely independent. ∗ A fortiori, the families (C h`(ξ), ` (ξ)i)i∈I are freely independent.

Suppose that K ⊂ H is an R-linear subspace with the property that hk1, k2i ∈ R for

every k1, k2 ∈ R. Then the von Neumann algebra generated by elements of the form s(ξ) := ∗ `(ξ) + ` (ξ) for ξ ∈ K is a II1 factor (provided dim(K) > 1, as otherwise it is abelian) with tracial state given by the vacuum state. 16 1.1.8 Operator-valued free probability.

We introduce brieﬂy the concept of operator-valued free probability, also called free probability with amalgamation. The reader is encouraged to consult any of [16,18,22] for a more in-depth treatment. The concepts here are meant to serve as an analogue to conditional probability in classical probability theory; this discussion will necessitate returning brieﬂy to the context of measure spaces, probability measures, and σ-algebras.

Suppose that (Ω, F0, P) is a probability space, and F ⊂ F0 is a sub-σ-algebra. A conditional expectation is a map

∞ ∞ E [·|F]: L (Ω, P) → L (Ω, P) with the properties that for any X ∈ L∞(Ω, P), E [X|F] is F-measurable and for all A ∈ F, Z Z X dP = E [X|F] dP. A A

We notice that the range of E [·|F] is a subalgebra of L∞(Ω, P) consisting of those functions which are F-measurable. It also follows that if Y ∈ L∞(Ω, P) is F-measurable, then E [XY |F] = E [X|F] Y for any X ∈ L∞(Ω, P). With these properties in hand, we are able to deﬁne a conditional expectation in the non-commutative setting.

Deﬁnition 1.1.16. Suppose that A is a ∗-algebra, and 1A ∈ B ⊂ A is a ∗-subalgebra. A conditional expectation is a linear map EB : A → B such that:

0 0 0 • for any b, b ∈ B and a ∈ A, EB (bab ) = bEB (a) b , and

• for any b ∈ B, EB(b) = b.

If such a map EB exists, the pair (A, EB) is said to be a B-valued probability space, and its elements are referred to as B-valued random variables.

Once again, one can do a free product construction of B-valued probability spaces. This leads to the following deﬁnition of amalgamated free independence.

17 Deﬁnition 1.1.17. Suppose (A, E) is a B-valued probability space, and for ι ∈ I, let (ι) (ι) A ⊂ A be subalgebras with B ⊂ A . Then (Aι)ι∈I are said to be freely independent with

(ij ) amalgamation over B if whenever a1, . . . , an ∈ A are such that aj ∈ A with i1 6= · · ·= 6 in and E(aj) = 0 for each j, we have E(a1 ··· an) = 0.

Example 1.1.18. Suppose that A(1), A(2) ⊂ A are freely independent in the tracial non- ∼ commutative probability space (A, τ). Let B = Mk(C) ⊂ Mk(C) ⊗ A = Mk(A), and (1) (2) E := 1 ⊗ τ. Then Mk(A ) and Mk(A ) are free with amalgamation over B. Indeed, suppose A1,...,An ∈ Mk(A) come from alternating subalgebras and satisfy E(Ai) = 0.

Then if Ai = (ai;j,k)j,k, we have τ(ai;j,k) = 0. But each entry in A1 ··· An is a sum of

(1) products of the form a1;j1,k1 a2;j2,k2 ··· an;jn,kn and since these come from alternating A , (2) A we ﬁnd τ(a1;j1,k1 ··· an;jn,kn ) = 0. Hence E(A1 ··· An) = 0.

1.1.9 Free entropy and regularity.

Free entropy was originally introduced by Voiculescu in [24] as an analogue of Shannon’s entropy from information theory. Roughly speaking, it may be thought of as giving a measure of how smoothly a tuple of non-commutative random variables is distributed.

If we are to speak of smoothly distributed variables, it would be helpful to know the free analogue of the Gaussian distribution.

Deﬁnition 1.1.19. Suppose that (A, ϕ) is a non-commutative probability space. A semicircular family in A is a tuple (S1,...,Sn) of self-adjoint elements of A such that for any n > 0 and i1, . . . , in,

κn(Si1 ,...,Sin ) = 0 unless n = 2.

The reason for the term “semicircular” is that the density of the random variable Si is given by 2 √ R2 − x2χ (x), πR2 [−R,R]

18 where R = 2κ2(Si,Si). A family of free semicircular variables may be realized on a Fock

n ∗ space: taking e1, . . . , en ∈ C to be the standard orthonormal basis, if Si = `(ei) + ` (ei) then (S1,...,Sn) are freely independent semicirculars with κ2(Si,Sj) = δi=j. Semicircular families take on the role of Gaussian variables from free probability: for example, central limit-type sums converge in distribution to semicircular families.

There are two approaches to free entropy theory; so far it is unknown whether or not they produce diﬀerent theories. The ﬁrst historically is the “microstates” version of free entropy, introduced and developed by Voiculescu in [24–27]. The entropy of a tuple of operators, χ(x1, . . . , xn), is related to the how many microstates the operator has among the N × N matrices in the limit as N tends to ∞, where a microstate is tuple of matrices whose distribution under the normalized trace is approximately that of the operators x1, . . . , xn.

We will not give a deﬁnition of microstates free entropy here, as it is complex and we will be more concerned with the “non-microstates” approach detailed below. However, we will state the following theorems from [25] which will be of great use to us, as they allow us to compute entropy directly in some cases.

∞ Theorem 1.1.20. Suppose that X ∈ L (Ω, P) is a self-adjoint random variable, and µX its law. Then 3 1 χ(X) = log |s − t| dµ (s) dµ (t) + + log 2π. x X X 4 2 R2 An immediate consequence of the above is that if X has any atoms in its spectral measure, χ(X) = −∞.

Theorem 1.1.21. Suppose that X1,...,Xn are self-adjoint and freely independent in the non-commutative probability space (M, τ). Then

n X χ(X1,...,Xn) = χ(Xi). i=1

1.1.9.1 Non-microstates free entropy.

Non-microstates free entropy was introduced by Voiculescu in [28] in an attempt to circum- vent the computational diﬃculties of the microstates version. The approach is to begin by 19 deﬁning a free analogue of Fisher information, and using an analogy with classical probability, taking it to be the derivative of the entropy function.

Deﬁnition 1.1.22. Suppose that X1,...,Xn ∈ M are self-adjoint elements in a ﬁnite tracial von Neumann algebra (M, τ) which are algebraically free (i.e., satisfy no polynomial identity).

Let A := C hX1,...,Xni ⊂ M be the algebra of polynomials in X1,...,Xn. Finally, suppose 2 that A generates M. Then for 1 ≤ i ≤ n, ∂i is the densely-deﬁned derivation from L (M) →

2 2 L (M) ⊗ L (M) with domain A, deﬁned by linearity, the Leibniz rule (i.e., ∂i(fg) = f ·

∂i(g) + ∂i(f) · g), and the condition

∂i(Xj) = δi=j1 ⊗ 1.

We pause here to introduce some notation: if X is an N-N-bimodule, given ξ ∈ X and a, b ∈ N we write (a ⊗ b)#ξ := a · ξ · b, and extend linearly to N ⊗ N.

Now, if n = 1, and one interprets C hXi ⊗ C hXi as polynomial functions from C2 → C, one ﬁnds that f(s) − f(t) ∂ (f) = . i s − t More generally, for arbitrary n, we have

n X ∂i (f)#(Xi ⊗ 1 − 1 ⊗ Xi) = f ⊗ 1 − 1 ⊗ f. i=1

For this reason, ∂i is often referred to as the free diﬀerence quotient. Also, notice that if

m = Xi1 ··· Xin is a monomial, then

X ∂i(m) = Xi1 ··· Xij−1 ⊗ Xij+1 ··· Xin .

j:ij =i

Deﬁnition 1.1.23. Suppose that X1,...,Xn are self-adjoint and generate a ﬁnite tracial

∗ von Neumann algebra (M, τ). Suppose further that 1 ⊗ 1 ∈ D(∂i ). Then the conjugate ∗ 2 variable of Xi is deﬁned to be ∂i (1 ⊗ 1) ∈ L (M).

Suppose that ξ1, . . . , ξn are the conjugate variables for X1,...,Xn. Then for any f ∈

C hX1,...,Xni, we have τ (ξif) = τ ⊗ τ (∂if) . 20 Deﬁnition 1.1.24. Suppose that X1,...,Xn are self-adjoint variables which generate a

ﬁnite tracial von Neumann algebra (M, τ). If the conjugate variables to X1,...,Xn exist

and are given by ξ1, . . . , ξn, then the (non-microstates) free Fisher information is deﬁned as

n ∗ X 2 Φ (X1,...,Xn) := kξik2 . i=1

∗ If any of the conjugate variables fails to exist, we set Φ (X1,...,Xn) := ∞.

We further deﬁne the (non-microstates) free entropy to be

Z ∞ ∗ 1 n ∗ 1 1 n χ (X1,...,Xn) := − Φ (X1 + t 2 S1,...,Xn + t 2 Sn) dt + log 2πe, 2 0 1 + t 2 where (S1,...,Sn) is a free semicircular system with variance 1, free from (X1,...,Xn).

Theorem 1.1.25. Suppose that X1,...,Xn are self-adjoint random variables in a ﬁnite tracial von Neumann algebra (M, τ). Then we have the following:

∗ • χ (X1) = χ(X1) (due to Voiculescu in [28]);

∗ • χ(X1,...,Xn) ≤ χ (X1,...,Xn) (due to Biane, Capitaine, and Guionnet in [5]).

Example 1.1.26. Suppose that S1,...,Sn are a free semicircular family with variance 1.

Then ξi = Si are the conjugate variables. This statement is established by showing the following identity holds for arbitrary f ∈ C hS1,...,Sni:

τ(Sif) = τ ⊗ τ(∂if ⊗ f).

The above identity can be thought of as the free version of the following identity which holds

for Gaussian variables X1,...,Xn:

0 E [Xif(X1,...,Xn)] = E [f (X1,...,Xn)] .

∗ Moreover, it follows that Φ (S1,...,Sn) = n.

A related concept is that of a dual system.

21 Deﬁnition 1.1.27. Suppose that X1,...,Xn are self-adjoint and generate a ﬁnite tracial von Neumann algebra (M, τ). A dual system to the operators is a tuple (Y1,...,Yn) of operators in B(L2(M)) such that

[Yi,Xj] = δi=jP1,

2 where P1 is the orthogonal projection onto 1 ∈ L (M). More generally, if M ⊂ B(H), we may replace P1 by any ﬁxed rank one projection.

∼ 2 Notice that with the identiﬁcation C hX1,...,Xni ⊗C hX1,...,Xni = FR(L (M)) given

by a ⊗ b 7→ aτ(b·), one has [Yi,T ] = (∂i(T ))#(1 ⊗ 1). One of the results in [28] is that existence of a dual system for a tuple of operators is stronger than the assumption of ﬁnite free ﬁsher information.

∗ n Example 1.1.28. Suppose that Si := `(ei) + ` (ei) ∈ B(F(C )), so that (S1,...,Sn) is a ∗ ∗ free semicircular family. Then the family admits a dual system given by (r (e1), . . . , r (en)).

∗ ∗ ∗ Indeed, [r (ei), ` (ej)] = 0 while [r (ei), `(ej)] = δi=jPΩ.

1.1.9.2 Algebraic variables.

Another useful sense of regularity of the distribution of a random variable is that of algebraicity.

Deﬁnition 1.1.29. Suppose that X ∈ L∞(Ω, P) is self-adjoint. Then X is said to be algebraic if the Cauchy transform of its distribution,

Z 1 SµX (z) := dµX (ζ), R z − ζ is algebraic as a formal power series in z.

22 CHAPTER 2

Bi-free independence.

The goal of this chapter is to make an examination of bi-free independence as introduced by Voiculescu in [30], and as introduced in Subsection 1.1.5. We will show that the combinatorial bi-free independence of Mastnak and Nica is equivalent to bi-free independence, and also develop a vanishing moment condition inspired by the one mentioned in Proposition 1.1.2. To do this, we will need to introduce some combinatorial machinery.

2.1 Some combinatorial structures.

We need two main combinatorial structures for our arguments: bi-non-crossing partitions, and shaded LR-diagrams.

2.1.1 Bi-non-crossing partitions.

−1 Deﬁnition 2.1.1. Let χ :[n] → {`, r}, and let sχ ∈ Sn be as above: if χ (`) =

−1 {i1 < ··· < ik} and χ (r) = {ik+1 > ··· > in}, then sχ(j) := ij. Then χ induces an or-

−1 −1 dering on [n] via i ≺χ j if and only if sχ (i) < sχ (j). We will also denote intervals in this ordering by [i, j]χ := {k ∈ [n]: i χ k χ j} (and use similar notation for open and half-open intervals), and refer to such sets as χ-intervals.

The set of bi-non-crossing partitions with respect to χ (or χ-non-crossing partitions) is equal to the set of non-crossing partitions on the ordered set {(i, χ(i)) : i ∈ [n]}, with the ordering given by ≺χ on the first coordinate. We denote the set of such partitions by BNC(χ). It will be convenient for us to think of elements of BNC(χ) as partitions of [n], and this is accomplished by projecting onto the first coordinate; from here on we will always make this 23 identification implicitly.

The reason we have formally make π include the information from χ is two-fold: we wish to be able to recover χ from π, and we wish BNC(χ) to be formally disjoint for diﬀerent χ. However, this technicality is almost never worth calling attention to. We also remark that χ-non-crossing partitions correspond precisely to those which may be drawn without crossings between a pair of vertical lines, with labelled nodes added to the left or right line

according to χ. In this picture, the ordering ≺χ corresponds to reading the labels down the left line and then up the right line. If we have such a diagram drawn using horizontal and vertical lines so that every block of π contains at most one vertical line segment, we will refer to that segment as the spine of the block it corresponds to, and the horizontal pieces as ribs.

Example 2.1.2. Suppose χ : [8] → {`, r} is such that χ−1(`) = {1, 3, 4, 6, 8} and χ−1(r) = {2, 5, 7}.

1 2 3 4 5 6 7 8 Then {{1, 3} , {2, 4} , {5, 6, 7, 8}} ∈ BN C(χ) even though it is not a non-crossing partition.

We also have the ordering 1 ≺χ 3 ≺χ 4 ≺χ 6 ≺χ 8 ≺χ 7 ≺χ 5 ≺χ 2.

−1 Finally, observe that BNC(χ) = π ∈ P(n): sχ · π ∈ N C(n) ; thus we have the following expression for the bi-free cumulants: X ϕ(z1 ··· zn) = κπ(z1, . . . , zn). π∈BN C(χ)

2.1.2 Shaded LR-diagrams.

We will now introduce a set of diagrams, called shaded LR-diagrams, which will be of use when computing joint moments of bi-free variables. In this context, a diagram consists of 24 the following data:

• n ∈ N0;

• a map χ :[n] → {`, r};

• a map ι :[n] → I;

• a χ-non-crossing partition π which in P(n) satisﬁes π ≺ {ι−1(j): j ∈ I}; and

• a subset S ⊂ π which are outer in the sense that if B ∈ S and C ∈ π, and there are

i, k ∈ C, j ∈ B with i ≺χ j ≺χ k, then B = C.

The diagrammatic representation will be given by drawing the partition π, colouring the blocks of π according to ι, and extending the spines of blocks in S to the top of the diagram. Note that the outerness assumption on the blocks in S ensures that this can be done in a non-crossing fashion; moreover, it allows us to order the blocks in S by ≺χ. We will often abbreviate the condition π ≺ {ι−1(j): j ∈ I} simply as π ≺ ι. Given B ∈ π, we will often write ι(B) as shorthand for the value ι(b) for b ∈ B.

Example 2.1.3. Let χ : [8] → {`, r} be as in Example 2.1.2, deﬁne ι by the sequence ( , , , , , , , ) , and let S = {{1, 3} , {2, 4}}. Then we have the following drawing of this diagram:

1 2 3 4 5 6 7 8

We now construct the set of shaded LR-diagrams recursively. First, we insist that the unique diagram with n = 0 (hence no nodes) is a shaded LR-diagram. Now, suppose D = (n, χ, ι, π, S) is a shaded LR-diagram, and letχ ˆ :[n + 1] → {`, r}, ˆι :[n + 1] → I be such that for i > 1,χ ˆ(i + 1) = χ(i) and ˆι(i + 1) = ι(i). We prescribeπ ˆ in the following way:

25 • (i + 1) ∼πˆ (j + 1) if and only if i ∼π j;

• if S = ∅, then 1 is a singleton inπ ˆ;

• if B ∈ S is the block with spine closest to theχ ˆ(1) side of the diagram and the colour

of B matches ˆι(1), then 1 ∼πˆ (j + 1) for every j ∈ B;

• if neither of the above conditions occurs, 1 is a singleton inπ ˆ.

Notice thatπ ˆ ∈ BN C(ˆχ) as π ∈ BN C(χ) and all blocks of S are outer; notice also that the block containing 1 inπ ˆ is also outer. Finally, let Sˆ be the set of blocks in S which, after relabelling, are still blocks ofπ ˆ; that is, if 1 was added to a pre-existing block of π, ˆ ˆ we exclude it from S. Then if B1 ∈ πˆ is the block containing 1, both (n + 1, χ,ˆ ˆι, π,ˆ S) and ˆ (n+1, χ,ˆ ˆι, π,ˆ S ∪{B1}) are shaded LR-diagrams. The set of shaded LR-diagrams, LR, is the smallest set which is closed under the above extension. Notice that each shaded LR-diagram on n+1 nodes arises through this construction from a unique LR-diagram on n nodes, which can be recovered simply by deleting the top part of the diagram.

We deﬁne some particular subsets of LR:

LRk := {(n, χ, ι, π, S) ∈ LR : |S| = k} ,

LR(χ, ι) := {(n, χ0, ι0, π, S) ∈ LR : χ0 = χ, ι0 = ι} ,

LRk(χ, ι) := LRk ∩ LR(χ, ι).

Example 2.1.4. Let D = (8, χ, ι, π, S) be as in Example 2.1.3. Suppose we wish to extend D with ˆι(1) = . Then we have the following four diagrams as options, depending on the choice ofχ ˆ(1).

1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7 7 8 8 8 8 9 9 9 9 26 The ﬁrst and last diagrams lie in LR2, the second lies in LR3, and the third lies in LR1.

Notice that we can only increase the number of spines reaching the top of the diagram when we add a node whose colour does not match that of the extended spine nearest it; thus the spines reaching the top of the diagram will always be of alternating colours.

Deﬁnition 2.1.5. Fix n ∈ N, and let χ :[n] → {`, r} and ι :[n] → I. We deﬁne the set BNC(χ, ι) as BNC(χ, ι) := {π ∈ BN C(χ):(n, χ, ι, π, ∅) ∈ LR} .

Note that even if ι is constant on the blocks of π ∈ BN C(χ), it is not necessarily true that π ∈ BN C(χ, ι)! This is because we have asked that π can be constructed through a certain process which connects blocks of the same colour when both are on the same level and such a connection maintains the bi-non-crossing property.

Example 2.1.6. If n = 3, ι ≡ , and χ ≡ `, π = {{1, 3} , {2}} ∈/ BN C(χ, ι).

1 2 3

If it were being constructed as an LR-diagram, it would be necessary to connect 2 to the spine extending upwards from 3 when it is added.

In order to better discuss these conditions, we introduce some terminology:

Deﬁnition 2.1.7. Let π ∈ BN C(χ), and B,C ∈ π. Then B and C are said to be piled if max(min(B), min(C)) ≤ min(max(B), max(C)). Diagrammatically, this means that there is necessarily some horizontal level on which both the spines of B and C are present.

A third block D ∈ π separates B from C if it is piled with both, equal to neither, and its spine lies between the spines of B and C. Equivalently, D is piled with B and C, and there are j, k ∈ D so that B ⊂ (j, k)χ while C ∩ [j, k]χ = ∅, or vice versa.

Finally, B and C are tangled if they are piled and no block separates them.

We extend these notions to LR in the obvious way. 27 We can now succinctly state the condition on partitions being in BNC(χ, ι): there must be no two tangled blocks of the same colour. We also remark that any element π ∈ BN C(χ) lies in BNC(χ, ι) for some ι :[n] → I provided |I| ≥ 2: one may first colour a block of π, then colour all blocks it is tangled with in the opposite colour, and continue in this fashion until π is coloured or there are no more restrictions, at which point another uncoloured block may be coloured. Since the relation of entanglement does not allow for any cycles (any given non-singleton block splits the diagram into two or more regions, and only it may be tangled with blocks in more than one of these regions), this will always produce a consistent colouring. We conclude [ BNC(χ) = BNC(χ, ι). ι∈In Definition 2.1.8. Suppose π, σ ∈ BN C(χ). We say σ is a lateral refinement of π and write

Example 2.1.9. Suppose χ = (`, r, `, r) and π = 1χ. Then {{1, 2} , {3, 4}}

{{1, 3} , {2, 4}}< 6 lat π.

1 1 2 2 3 3 4 4

2.2 Computing joint moments of bi-free variables.

We will now make explicit the connection between shaded LR-diagrams and joint moments (ι) ˚(ι) (ι) of bi-free variables. Let (V , V , ξ )ι∈I be a family of vector spaces with speciﬁed state (ι) (ι) (ι) ˚ vectors, and let π` , πr be the left and right representations of L(V ) on (V, V , ξ) := (ι) ˚(ι) (ι) (ι) (ι) ˚(ι) (ι) (ι) ∗ι∈I (V , V , ξ ). Also, deﬁne ψ : V → C as the functional with v ∈ V + ψ ξ , and let p(ι)(x) = ψ(ι)(x)ξ(ι), and ψ, p in a similar fashion for V . Notice that, for T ∈ L(V (ι)),

(ι) (ι) (ι) p T p = ϕξ(ι) (T )p .

Let n ∈ N. We deﬁne a map Ξ which takes as arguments B ⊂ [n] and T1,...,Tn ∈ 28 S (ι) ι∈I L(V ) with the requirement that all operators corresponding to elements of B come ( (θ) (θ) from a single L V ), and produces a vector in V as follows: with B = k1 < ··· < k|B| ,

(θ) (θ) (θ) (θ) Ξ(B; T1,...,Tn) := Tk1 (1 − p )Tk2 (1 − p )Tk3 ··· (1 − p )Tk|B| ξ .

S (ι) Given a map ι :[n] → I, we say a sequence of operators T1,...,Tn ∈ ι∈I L(V ) (ι) is consistent with ι if Ti ∈ L(V ) for each i ∈ [n]. We will now deﬁne a map Ψ which takes as arguments a shaded LR-diagram D = (n, χ, ι, π, S) and a sequence of operators

T1,...,Tn consistent with ι, and produces an element of V . Let S = {B1,...,Bk} with

B1 ≺χ · · · ≺χ Bk. Then

  k ! Y (ι) O (ι(Bi)) Ψ(D; T1,...,Tn) :=  ψ (Ξ(B; T1,...,Tn)) (1 − p )Ξ(Bi; T1,...,Tn) . B∈π\S i=1

Here the tensor product should be interpreted as ordered from least i to largest, and if k = 0, the empty tensor product factor should be read as the vector ξ.

Example 2.2.1. Let D = (8, χ, ι, π, S) be as in Example 2.1.3, and take T1,...,T8 consistent with ι. Then

( ) ( ) ( ) ( ) ( ) Ψ(D; T1,...,T8) = ψ T5(1 − p )T6(1 − p )T7(1 − p )T8ξ

( ) ( ) ( ) ( ) ( ) ( ) · (1 − p )T1(1 − p )T3ξ ⊗ (1 − p )T2(1 − p )T4ξ .

Proposition 2.2.2. Fix χ :[n] → {`, r} and ι :[n] → I. Let T1,...,Tn be consistent with ι. Then (ι(1)) (ι(n)) X πχ(1) (T1) ··· πχ(n) (Tn)ξ = Ψ(D; T1,...,Tn). D∈LR(χ,ι) Moreover,  

(ι(1)) (ι(n)) X  X |π|−|σ| ϕ(π (T1) ··· π (Tn)) =  (−1)  ϕσ(T1,...,Tn). χ(1) χ(n)   σ∈BN C(χ) π∈BN C(χ,ι) σ≤latπ

Proof. We will establish the ﬁrst identity by induction on n. The case n = 0 is immediate since if D is the empty diagram, Ξ(D; ) = ξ. Therefore suppose n > 0; our induction

29 hypothesis applied to T2,...,Tn tells us that, withχ ˆ(j) = χ(j + 1) and ˆι(j) = ι(j + 1) for 1 ≤ j < n, (ι(2)) (ι(n)) X πχ(2) (T2) ··· πχ(n) ξ = Ψ(D; T2,...,Tn). D∈LR(ˆχ,ˆι) Fix D = (n − 1, χ,ˆ ˆι, π, S) ∈ LR(ˆχ, ˆι), and suppose for the moment that χ(1) = `.

Let D0,D1 ∈ BN C(χ, ι) be the two diagrams constructed from D, where D1 has a spine extending from 1 to the top and D0 does not.

(j) Recall the bijections Wj : V ⊗ V (`, j) → V from Subsection 1.1.2. If S is empty or its ≺χ-least element (i.e., the left-most spine reaching the top) is B with ι(B) 6= ι(1), then w := Ψ(D; T2,...,Tn) ∈ V (`, ι(1)); hence

(ι(1)) (ι(1)) π` (T1)w = Wj (T1 ⊗ 1)(ξ ⊗ w)

(ι(1)) (ι(1)) (ι(1)) = ψ (T1)w + (1 − p )T1ξ ⊗ w

(ι(1)) (ι(1)) = ψ (Ξ({1} ; T1,...,Tn)) w + (1 − p )Ξ({1} ; T1,...,Tn) ⊗ w

= Ψ(D0; T1,...,Tn) + Ψ(D1; T1,...,Tn).

ˆ ˆ On the other hand, suppose the ≺χ-least element of S is B with ι(B) = ι(1); let B = n o ˆ ˆ ˚(ι(1)) {1} ∪ j + 1 : j ∈ B . Let v := Ξ(B; T2,...,Tn) ∈ V , and let w ∈ V (`, ι(1)) be so that ˚ (ι(1)) W(ι(1))(v ⊗ w) = Ψ(D; T2,...,Tn). Since v ∈ V (ι(1)), we have (1 − p )v = v. Then we have

(ι(1)) π` (T1)Ψ(D; T2,...,Tn) = Wj ((T1 ⊗ 1)(v ⊗ w))

(ι(1)) (ι(1)) (ι(1)) (ι(1)) = ψ (T1(1 − p )v)w + (1 − p )T1(1 − p )v ⊗ w

(ι(1)) (ι(1)) = ψ (Ξ(B; T1,...,Tn)) w + (1 − p )Ξ(B; T1,...,Tn) ⊗ w

= Ψ(D0; T1,...,Tn) + Ψ(D1; T1,...,Tn).

As each D0 ∈ LR(χ, ι) arises from exactly one diagram in LR(ˆχ, ˆι), we have

(ι(1)) (ι(n)) X π` (T1) ··· πχ(n) ξ = Ψ(D; T1,...,Tn). D∈LR(χ,ι) The argument for χ(1) = r is the same, except we use the decomposition V ∼= V (r, ι(1)) ⊗

(ι(1)) V and work with the ≺χ-greatest element of S (i.e., the furthest right spine which reaches the top). 30 We now must establish the second claimed identity, which will follow from the ﬁrst. Indeed, notice that

(ι(1)) (ι(n)) (ι(1)) (ι(n)) ϕ(πχ(1) (T1) ··· πχ(n) (Tn)) = ψ(πχ(1) (T1) ··· πχ(n) (Tn)ξ).

Applying ψ to the ﬁrst equation, we see that the only terms on the right hand side which

survive correspond to diagrams in LR0(χ, ι).

Let D = (n, χ, ι, π, ∅) ∈ LR0(χ, ι) be such a diagram, and take B = {k1 < ··· < kb} ∈ π. Then

(ι(B)) ψ (Ξ(B; T1,...,Tn))

(ι(B)) (θ) (θ) (θ) (θ) = ψ Tk1 (1 − p )Tk2 (1 − p )Tk3 ··· (1 − p )Tk|B| ξ

X X m = (−1) ϕ (ι(B)) T ··· T ··· ϕ (ι(B)) T ··· T ξ ξ k1 kq1 ξ kqm+1 kb m≥0 1≤q1<···

Each term in the sum corresponds to a lateral reﬁnement of π with cuts only in the block B, with sign determined by the parity of the number of cuts. Then there is a correspondence between lateral reﬁnements of π and terms in ! Y X X m (−1) ϕ (ι(B)) T ··· T ··· ϕ (ι(B)) T ··· T ξ , ξ k1 kq1 ξ kqm+1 kb B∈π m≥0 1≤q1<···

ψ (Ψ(D; T1,...,Tn)) ! Y X X m = (−1) ϕ (ι(B)) T ··· T ··· ϕ (ι(B)) T ··· T ξ ξ k1 kq1 ξ kqm+1 kb B∈π m≥0 1≤q1<···

X |π|−|σ| = (−1) ϕσ(T1,...,Tn). σ∈BN C(χ) σ≤latπ

Summing over D ∈ LR0(χ, ι) (or equivalently, π ∈ BN C(χ, ι)) and reversing the order of the summations yields the desired identity.

(ι) (ι) Corollary 2.2.3. Let (A` , Ar ) be a family of pairs of faces in a non-commutative ι∈I (ι) (ι) probability space (A, ϕ). Then (A` , Ar ) are bi-freely independent if and only if for ι∈I 31 (ι(i)) every n ∈ N, χ :[n] → {`, r}, ι :[n] → I, and z1, . . . , zn ∈ A with zi ∈ Aχ(i) , we have  

X  X |π|−|σ| ϕ(z1 ··· zn) =  (−1)  ϕσ(z1, . . . , zn).   σ∈BN C(χ) π∈BN C(χ,ι) σ≤latπ

(ι) (ι) Proof. If (A` , Ar ) are bi-free, this follows from applying the previous proposition to ι∈I the representation guaranteed by the deﬁnition of bi-freeness.

Conversely, suppose that the above equation holds whenever appropriate. Let

(ι) (ι) (ι) µ = ∗∗ ϕ : ∗ A` ∗ Ar → ι∈I ι∈I C

(ι) (ι) (ι) (ι) be the state with respect to which (A` , Ar ) are bi-free in ∗ι∈I A` ∗ Ar such that ι∈I (ι) µ| (ι) (ι) = ϕ . Then we have A` ∗ Ar  

X  X |π|−|σ| ϕ(z1 ··· zn) =  (−1)  ϕσ(z1, . . . , zn)   σ∈BN C(χ) π∈BN C(χ,ι) σ≤latπ  

X  X |π|−|σ| =  (−1)  µσ(z1, . . . , zn)   σ∈BN C(χ) π∈BN C(χ,ι) σ≤latπ

= µ(z1 ··· zn).

(ι) (ι) But then (A` , Ar ) are bi-free with respect to ϕ since they are bi-free with respect to ι∈I µ.

2.3 The incident algebra on bi-non-crossing partitions.

The main goal of this section is to develop the appropriate M¨obiusfunction for bi-non- crossing partitions, with an eye towards applying it to the bi-free cumulant functionals.

Deﬁnition 2.3.1. The incidence algebra on the lattice of bi-non-crossing partitions consists of the following set of functions:  

 a a  IA(BNC) := f : BNC(χ) × BN C(χ) → C f(σ, π) = 0 unless σ ≤ π .

 n≥1 χ:[n]→{`,r}  32 Once again, we deﬁne the convolution product: X (f ? g)(σ, π) := f(σ, ρ)g(ρ, π). σ≤ρ≤π

It is immediately veriﬁed that the convolution is associative, essentially because X X X X X = = . σ≤ρ≤π ρ≤ρ0≤π σ≤ρ≤ρ0≤π σ≤ρ0≤π σ≤ρ≤ρ0

2.3.1 Multiplicative functions.

We wish to demonstrate that the same kind of decomposition into products of full intervals holds in the bi-non-crossing case as in the non-crossing case, so that we can make sense of multiplicative functions. However, we want this decomposition to be aware of more than just the lattice structure; it should be consistent with the choice of sides. We adapt the decomposition of Proposition 1.1.7 from the free setting as follows.

Suppose that we have σ ≤ π ∈ BN C(χ). We ﬁrst split both partitions according to the blocks of π: Y [σ, π] −→ [σ|B, {B}], B∈π

where σ|B and {B} are thought of as elements of BNC(χ|B); thus we will from here on

assume π = 1χ.

Now, if possible, let B = {i1 ≺χ · · · ≺χ ik} be a block of σ so that there is some a ∈ [n]

0 and 1 ≤ j < k so that ij ≺χ a ≺χ ij+1. this case, let I1 = (ij, ij+1)≺χ , I0 = [n] \ I1, and 0 0 I1 = I1 ∪ {max(ij, ij+1)}; then take σ0 and σ1 to be the sub-partitions of σ corresponding to 0 0 the sets I0 and I1, and σ1 = σ1 ∪ {max(ij, ij+1)}. We then have h i h i [σ, 1 ] −→ σ , 1 × σ , 1 . χ 0 χ|I0 1 χ|I1

If σ = 1 , we exclude the left hand term above. 0 χ|I0 Lastly, if σ consists entirely of χ-intervals, we retract each interval to its lowest node.

0 That is, if σ = {B1,...,Bm}, ij = max(Bj), and i1 < ··· < im, then taking χ (j) = χ(ij) we have

[σ, 1χ] −→ [0χ0 , 1χ0 ]. 33 Example 2.3.2. Letting a diagram below stand for the interval between it and the corresponding maximum partition,

∼= × ∼= ×

[0(r,r), 1(r,r)] × [0(`,r,r), 1(`,r,r)]

Proposition 2.3.3. The decomposition described above provides an isomorphism of lattices.

∼ Proof. We note that as lattices, [σ, π] = [sχ · σ, sχ · π]. After performing this identiﬁca- tion, every step of the decomposition above corresponds to one in the decomposition of

[sχ · σ, sχ · π] in the lattice of non-crossing partitions, as in [20, Proposition 1] (restated in Proposition 1.1.7). All we have done is make choices of sides for the new nodes which are added, and for nodes obtained by contracting blocks, but we have done so in such a way as to make them consistent with the original lattice.

Deﬁnition 2.3.4. A function f ∈ IA(BNC) is said to be multiplicative if whenever σ, π ∈ BNC(χ) are such that [σ, π] decomposes as

m Y BNC(χj), j=1 it follows that m Y f(σ, π) = f(0χj , 1χj ). j=1 Lemma 2.3.5. The convolution of multiplicative functions is multiplicative.

The proof is identical to that of Proposition 1.1.10.

Deﬁnition 2.3.6. Mirroring the free case, we deﬁne some particular elements of IA(BNC):    1 if σ = π  1 if σ ≤ π δBNC(σ, π) := , and ζBNC(σ, π) := .  0 otherwise  0 otherwise

We further deﬁne the M¨obiusfunction µBNC by the relation µBNC ? ζBNC = δBNC. 34 To check that ζBNC is actually invertible, one may appeal to the fact that ζNC is. Indeed,

−1 −1 if σ, π ∈ BN C(χ), then ζBNC(σ, π) = ζNC(sχ ·σ, sχ ·π) (as the lattice BNC(χ) is isomorphic −1 −1 to NC(n), with isomorphism given by sχ ). Hence if one deﬁnes µBNC(σ, π) := µNC(sχ · −1 σ, sχ · π) it follows that µBNC ? ζBNC = δBNC = ζBNC ? µBNC.

Further, notice that δBNC is the convolutive identity, and δBNC and ζBNC are clearly multiplicative. We also have µBNC is multiplicative, and this may be veriﬁed by once again appealing to its relation to µNC.

Remark 2.3.7. Suppose that a`, ar ∈ A are elements in a non-commutative probability space. Let m, k ∈ IA(BNC) be the unique multiplicative functions with m(0χ, 1χ) =

ϕ(aχ(1) ··· aχ(n)) and k(0χ, 1χ) = κχ(aχ(1), . . . , aχ(n)). Then we once again have the relation m = k ? ζBNC and k = m ? µBNC. Likewise, we once again have

X κπ(a1, . . . , an) = ϕσ(a1, . . . , an)µBNC(σ, π). σ∈BN C(χ) σ≤π

2.4 Unifying bi-free independence.

We are now ready to start the main thrust of our argument to relate the concepts of bi-free independence and combinatorial bi-free independence.

2.4.1 Summation considerations.

Our ﬁrst goal is to put the condition from Corollary 2.2.3 into a more convenient form. Having introduced the M¨obiusfunction for the lattice BNC, we are now able to do so.

Proposition 2.4.1. Let χ :[n] → {`, r} and ι :[n] → I. Then for every σ ∈ BN C(χ) such that σ ≤ ι,

X |π|−|σ| X (−1) = µBNC(σ, π). π∈BN C(χ,ι) π∈BN C(χ) σ≤latπ σ≤π≤ι

Our strategy is to ﬁrst establish a simple case by appealing to free probability, and then reduce all other cases to it. 35 Lemma 2.4.2. The formula in Proposition 2.4.1 holds when χ ≡ `.

Proof. Notice that if χ ≡ `, we have BNC(χ) = NC(n) and

< ++ > µBNC|BNC(χ)×BN C(χ) = µNC|NC(n)×N C(n).

(ι) (ι(i)) Now if (A` )ι∈I are taken to be free, we have for any z1, . . . , zn with zi ∈ A` that X ϕ(z1 ··· zn) = κπ(z1, . . . , zn) π∈N C(n) X = κπ(z1, . . . , zn) π∈N C(n) π≤ι X X = µNC(σ, π)ϕσ(z1, . . . , zn) π∈N C(n) σ∈N C(n) π≤ι σ≤π   X  X  =  µNC(σ, π) ϕσ(z1, . . . , zn).   σ∈N C(n) π∈N C(n) σ≤ι σ≤π≤ι

(ι) On the other hand, (A` , C)ι∈I are bi-free, so we ﬁnd  

X  X |π|−|σ| ϕ(z1 ··· zn) =  (−1)  ϕσ(z1, . . . , zn)   σ∈BN C(χ) π∈BN C(χ,ι) σ≤latπ  

X  X |π|−|σ| =  (−1)  ϕσ(z1, . . . , zn).   σ∈N C(n) π∈BN C(χ,ι) σ≤latπ

Now, ﬁx π ∈ N C(n) with π ≤ ι, and construct variables y1, . . . , yn so that the appropriate collections are free and the only pure moments of these variables which do not vanish are those precisely corresponding to the blocks of π. Then only the term corresponding to π will survive in each of the above sums, and we conclude

X X X |π|−|σ| µBNC(σ, π) = µNC(σ, π) = (−1) . π∈BN C(χ) π∈N C(n) π∈BN C(χ,ι) σ≤π≤ι σ≤π≤ι σ≤latπ 36 Lemma 2.4.3. Let χ, ~χ :[n] → {`, r} be so that χ(j) = ~χ(j) for 1 ≤ j < n, χ(n) = `, and ~χ(n) = r. Take ι :[n] → I and let σ ∈ BN C(χ) be such that σ ≤ ι; let ~σ ∈ BN C (~χ) be the ~χ-non-crossing partition with the same blocks as σ. Then

X X (−1)|π|−|σ| = (−1)|~π|−|~σ|, π∈BN C(χ,ι) ~π∈BN C(~χ,ι) σ≤latπ ~σ≤lat~π and X X µBNC(σ, π) = µBNC(~σ, ~π). π∈BN C(χ) ~π∈BN C(~χ) σ≤π≤ι ~σ≤~π≤ι

Proof. Notice that there is a correspondence between BNC(χ) and BNC(~χ) given by identifying elements which have the same block structure, and moreover this also holds for BNC(χ, ι) and BNC(~χ,ι). Moreover, this identiﬁcation preserves the partial orderings ≤

and ≤lat, as well as µBNC. Thus both identities hold.

Lemma 2.4.4. Let χ :[n] → {`, r} and k ∈ [n − 1] be so that χ(k) = ` and χ(k + 1) = r. Take ι :[n] → I and let σ ∈ BN C(χ) be such that σ ≤ ι. Now, deﬁne χ0 :[n] → {`, r} and ι0 :[n] → I as follows:    χ(k + 1) if j = k  ι(k + 1) if j = k   χ0(j) = χ(k) if j = k + 1 , and ι0(j) = ι(k) if j = k + 1 .    χ(j) otherwise  ι(j) otherwise

Let σ0 ∈ BN C(χ0) be the bi-non-crossing partition obtained by interchanging k and k + 1 in σ (note that σ0 ≤ ι0). Then

X X 0 0 (−1)|π|−|σ| = (−1)|π |−|σ |, π∈BN C(χ,ι) π0∈BN C(χ0,ι0) 0 0 σ≤latπ σ ≤latπ and

X X 0 0 µBNC(σ, π) = µBNC(σ , π ). π∈BN C(χ) π0∈BN C(χ0) σ≤π≤ι σ0≤π0≤ι0

37 Proof. Once again, observe that the correspondence between BNC(χ) and BNC(χ0) obtained

by exchanging k and k + 1 is a bijection which preserves the partial orders ≤ and ≤lat, and

0 0 leaves µBNC invariant; this follows because sχ = sχ0 . Moreover, π ≤ ι if and only if π ≤ ι , so the second claimed equality holds. However, it is not necessarily the case that the bijection takes BNC(χ, ι) to BNC(χ0, ι0), as it may take a partition π ∈ BN C(χ) which has no tangled blocks of the same colour to one which does, or vice versa.

0 0 Note that this correspondence only fails when either exactly one of σ ≤lat π and σ ≤lat π holds (which requires that k σ k + 1 and k ∼π k + 1, which ensures ι(k) = ι(k + 1)) or exactly one of π ∈ BN C(χ, ι) and π0 ∈ BN C(χ0, ι0) holds (which requires that blocks of the

0 same colour are entangled in exactly one of π and π , so k π k + 1 but ι(k) = ι(k + 1)).

Thus the two sums agree unless k σ k + 1 and ι(k) = ι(k + 1). Therefore let us assume we are in this case.

We will split the sum on the left hand side of the ﬁrst equation into three sums: one over

partitions where k and k + 1 are in separated blocks of π; one over terms with k ∼π k + 1; and one over the rest.     X |π|−|σ|  X X X  |π|−|σ| (−1) =  + +  (−1) .   π∈BN C(χ,ι)  π∈BN C(χ,ι) π∈BN C(χ,ι) π∈BN C(χ,ι)  σ≤latπ σ≤latπ σ≤latπ σ≤latπ k,k+1 separated in π k∼πk+1 kπk+1 unseparated in π We now claim that the second and third sums cancel. Indeed, if k and k + 1 are in blocks which are distinct and not separated in π, then it must be the case that π ≤ {[1, k], [k + 1, n]}: k may not be connected to a lower node, nor k + 1 to a higher, or else they would be tangled and of the same colour; meanwhile, no other block may have nodes both less than and greater than k, or else it would separate the two. In particular, the partition ρ obtained by joining

the blocks containing k and k + 1 is still an element of BNC(χ, ι) and still has σ ≤lat ρ as the only additional cut required is a lateral one between k and k + 1. Further, it is clear that |ρ| = |π| − 1 so their contributions to the sums cancel. As this correspondence is easily

38 inverted, the contributions of the two sums vanish, and we are left with

X X (−1)|π|−|σ| = (−1)|π|−|σ|. π∈BN C(χ,ι) π∈BN C(χ,ι) σ≤latπ σ≤latπ k,k+1 separated in π

But now if π ∈ BN C(χ, ι) is such that k and k + 1 are in distinct separated blocks of π, then k and k+1 are in distinct separated blocks of π0 and in particular none of the conditions

0 0 we are concerned with can fail: σ ≤lat π as no cuts are necessary between k and k + 1, which is the only region where changes have been made; and no blocks have become entangled which were not before as the only blocks which have changed are separated. Likewise every π0 ∈ BN C(χ0, ι0) with k and k + 1 in separated blocks comes from such a π ∈ BN C(χ, ι). We conclude

X X (−1)|π|−|σ| = (−1)|π|−|σ| π∈BN C(χ,ι) π∈BN C(χ,ι) σ≤latπ σ≤latπ k,k+1 separated in π X 0 0 X 0 0 = (−1)|π |−|σ | = (−1)|π |−|σ |. π0∈BN C(χ0,ι0) π0∈BN C(χ0,ι0) 0 0 0 0 σ ≤latπ σ ≤latπ k,k+1 separated in π0

Example 2.4.5. We take a moment here to show that the correspondence in the proof above can indeed fail to carry BNC(χ, ι) onto BNC(χ0, ι0), and so our proof is not more complicated than necessary. Let χ be given by the sequence (`, `, r) and ι ≡ 1. Then χ0 corresponds to the sequence (`, r, `). Yet if σ = {{1, 2} , {3}} then σ0 = {{1, 3} , {2}}, and we have σ ∈ BN C(χ, ι) while σ0 ∈/ BN C(χ0, ι0): the issue is that in σ0, {2} is tangled with {1, 3}, even though {1, 2} and {3} were not tangled in σ.

1 1 2 2 3 3 σ σ0

It is likewise easy to obtain examples of σ ∈ BN C(χ) \BNC(χ, ι) with σ0 ∈ BN C(χ0, ι0).

39 Proof of Proposition 2.4.1. By Lemma 2.4.2, we have that the equation holds whenever χ ≡ `. By repeatedly applying Lemmata 2.4.2 and 2.4.3, we conclude that holds for all other choices of χ as well.

(ι) (ι) Corollary 2.4.6. Let (A` , Ar ) be a family of pairs of faces in a non-commutative ι∈I (ι) (ι) probability space (A, ϕ). Then (A` , Ar ) are bi-freely independent if and only if for ι∈I (ι(i)) every n ∈ N, χ :[n] → {`, r}, ι :[n] → I, and z1, . . . , zn ∈ A with zi ∈ Aχ(i) , we have   X  X  ϕ(z1 ··· zn) =  µBNC(σ, π) ϕσ(z1, . . . , zn).   σ∈BN C(χ) π∈BN C(χ) σ≤π≤ι

Proof. This follows immediately from Proposition 2.4.1 and Corollary 2.2.3.

2.4.2 Bi-free independence is equivalent to combinatorial bi-free independence.

(ι) (ι) Theorem 2.4.7. Let (A` , Ar ) be a family of pairs of faces in a non-commutative ι∈I (ι) (ι) probability space (A, ϕ). Then (A` , Ar ) are bi-freely independent if and only if they ι∈I are combinatorially bi-freely independent.

(ι) (ι) Proof. Suppose (A` , Ar ) are bi-free. We will show that all mixed bi-free cumulants ι∈I vanish by induction on n, with the case n = 1 being vacuously satisﬁed. Therefore ﬁx n ∈ N and χ :[n] → {`, r}, and let ι :[n] → I be non-constant; we assume that all mixed (ι(i)) cumulants with fewer than n arguments vanish. Take z1, . . . , zn ∈ A so that zi ∈ Aχ(i) . By Corollary 2.4.6, we have   X  X  ϕ(z1 ··· zn) =  µBNC(σ, π) ϕσ(z1, . . . , zn)   σ∈BN C(χ) π∈BN C(χ) σ≤π≤ι X X = µBNC(σ, π)ϕσ(z1, . . . , zn) π∈BN C(χ) σ∈BN C(χ) π≤ι σ≤π X = κπ(z1, . . . , zn). π∈BN C(χ) π≤ι

40 On the other hand, using the moment-cumulant formula we have

X ϕ(z1 ··· zn) = κπ(z1, . . . , zn). π∈BN C(χ)

But now any π ∈ BN C(χ) with π ι and |π| > 1 is a product of the cumulants corresponding

to its blocks, at least one of which must be mixed, and so κπ(z1, . . . , zn) = 0. Hence

X ϕ(z1 ··· zn) = κ1χ (z1, . . . , zn) + κπ(z1, . . . , zn) = κ1χ (z1, . . . , zn) + ϕ(z1 ··· zn). π∈BN C(χ) π≤ι

We conclude that κ1χ (z1, . . . , zn) = 0.

(ι) (ι) Now, for the other direction, suppose (A` , Ar ) are combinatorially bi-free. Once ι∈I again, let n ∈ N, χ :[n] → {`, r}, and ι :[n] → I. Using successively the moment-cumulant formula, combinatorial bi-free independence, and the formula for computing cumulants from Remark 2.3.7, we ﬁnd:

X X ϕ(z1 ··· zn) = κπ(z1, . . . , zn) = κπ(z1, . . . , zn) π∈BN C(χ) π∈BN C(χ) π≤ι X X = µBNC(σ, π)ϕσ(z1, . . . , zn) π∈BN C(χ) σ∈BN C(χ) π≤ι σ≤π   X  X  =  µBNC(σ, π) ϕσ(z1, . . . , zn).   σ∈BN C(χ) π∈BN C(χ) σ≤π≤ι

(ι) (ι) Then by Corollary 2.4.6, (A` , Ar ) are bi-freely independent. ι∈I

2.4.3 Voiculescu’s universal bi-free polynomials.

In [30, Section 5], Voiculescu provided a non-constructive proof of universal polynomials characterising bi-free independence; using Theorem 2.4.7 we are able to produce them con- cretely.

(ι) (ι) We will ﬁrst introduce some convenient notation. Suppose (zi )i∈I , (zj )j∈J are ι∈I pairs of two-faced families of non-commutative random variables, and suppose α :[n] → 41 I ` J, ι :[n] → I. Then we write

(1) (1) (1) ι (ι(1)) (ι(n)) ϕα(z ) := ϕ(zα(1) ··· zα(n)) and ϕα(z ) := ϕ(zα(1) ··· zα(n) ).

We also denote by χα :[n] → {`, r} the map such that χ(k) = ` if and only if α(k) ∈ I.

` Proposition 2.4.8. For each ι :[n] → I and α :[n] → I J we deﬁne the polynomial Pα,ι (ι) on indeterminates XK indexed by non-empty subsets K ⊂ [n] by the formula  

X  X  Y ι(V ) Pα,ι :=  µBNC(σ, π) X .   V σ∈BN C(χα,ι) π∈BN C(χα) V ∈σ σ≤π≤ι

(κ) (κ) Then for (zi )i∈I , (zj )j∈J a bi-free collection of two-faced families in (A, ϕ) we have κ∈I

ι (κ) ϕα(z ) = Pα,ι((z )κ∈I ) where P ((z(κ)) ) is given by evaluating P at Xκ = ϕ(zκ ··· zκ ) for κ ∈ I. α,ι κ∈I α,ι {k1<···

X Qα := Pα,ι, ι:[n]→I then (1) (2) X Qα = X[n] + X[n] + Pα,ι, ι:[n]→I ι non-constant and

(1) (2) ϕα(z) = Qα(z , z )

(1) (2) where Qα(z , z ) is Qα evaluated at the same point as the Pα,ι above.

ι Proof. The claim that ϕα(z ) may be computed by evaluating Pα,ι at the correct point is the

(1) (2) content of Corollary 2.4.6. The claim that ϕα(z) = Qα(z , z ) is obtained by expanding

(1) (2) (1) (2) X ι ϕα(z) = ϕ (zα(1) + zα(1)) ··· (zα(n) + zα(n)) = ϕα(z ), ι:[n]→I and then applying the first claim. 42 To establish the final piece of the proposition, that Qα has the presentation claimed, it (ι) suffices to show that Pα,ι = X[n] when ι is constant. But in this case,

X X µBNC(σ, π) = µBNC(σ, π) = δBNC(σ, 1),

π∈BN C(χα) π∈BN C(χα) σ≤π≤ι

(ι) and we have that the only term surviving in Pα,ι is X[n] .

` Proposition 2.4.9. For α :[n] → I J, recursively deﬁne polynomials Rα on indetermi-

nates XK indexed by non-empty subsets K ⊆ [n] by the formula

X Y Rα = µBNC(π, 1) XV .

π∈BNC(χα) V ∈π

If XK is given degree |K|, then Rα is homogeneous with degree n.

Given a two-faced family ((zi)i∈I , (zj)j∈J ) in a non-commutative probability space and

setting Rα(z) to be Rα evaluated with X{k1<···

Proof. The identity for Rα follows directly from the formula in Remark 2.3.7. The cumulant property then follows since κ possesses it.

The polynomials Pα,ι, Qα, and Rα are precisely the universal polynomials from Proposi- tions 2.18, 5.2, 5.6, and 5.7 of [30].

2.5 An alternating moment condition for bi-free independence.

So far we have still not presented a characterisation of bi-free independence as straightforward as the characterisation of free independence mentioned in Proposition 1.1.2: that families are free if and only if alternating products of centred variables are centred. In this section we will develop an analogous condition for bi-free independence.

43 (ι) (ι) Deﬁnition 2.5.1. Let (A` , Ar ) be a family of pairs of faces in a non-commutative ι∈I probability space (A, ϕ). We say the family has the vanishing alternating centred χ-interval Eigenschaft (which we will abbreviate as vaccine) if whenever:

• n ≥ 1, χ :[n] → {`, r}, ι :[n] → I, and

• z1, . . . , zn ∈ A are such that:

(ι(i)) – zi ∈ Aχ(i) ; and

– whenever {i1 < ··· < ik} is a maximal ι-monochromatic χ-interval, ϕ(zi1 ··· zik ) = 0,

it follows that ϕ(z1 ··· zn) = 0.

Example 2.5.2. Suppose χ is such that χ−1(`) = {2, 5, 6, 7, 8}, χ−1(r) = {1, 3, 4, 9, 10}, and ι corresponds to the colouring below (i.e., ι−1 ( ) = {1, 2, 3, 5, 8, 9, 10} and ι−1 ( ) = {4, 6, 7}).

1 2 3 4 5 6 7 8 9 10

The maximal ι-monochromatic χ-intervals are {2, 5} , {6, 7} , {8, 9, 10} , {4}, and {1, 3}. Vac-

cine would imply ϕ(z1 ··· z10) = 0 whenever z1, . . . , z10 are chosen corresponding to χ and ι with

0 = ϕ(z2z5) = ϕ(z6z7) = ϕ(z8z9z10) = ϕ(z4) = ϕ(z1z3).

The reason this condition becomes more complicated than in the free case amounts to the

fact that we cannot replace z1z3 or z8z9z10 with single elements of either the left or right faces; in the latter case, because neither face may contain an appropriate operator, and in

the former because z2 may not commute with z3 or z1 and so the two may not be moved next to each other. 44 2.5.1 The equivalence.

(ι) (ι) Lemma 2.5.3. Let (A` , Ar ) be a family of pairs of faces in a non-commutative ι∈I probability space (A, ϕ). Then the family has vaccine if the pairs of faces are bi-free.

Proof. Let n ≥ 1, χ :[n] → {`, r}, and ι :[n] → I, and denote by J the set of maximal ι-monochromatic χ-intervals in [n]. Note that J ∈ BNC(χ) may be thought of as a χ-non- crossing partition in its own right, which will sometimes be of use notationally. For P ⊂ [n], let m(P ) denote the minimum element of BNC(χ) containing P as a block, so all blocks of m(P ) except P are singletons. Let b : {π ∈ BNC(χ): π ≤ ι} → J be a function with the following properties:

• if π ∈ BNC(χ), j ∈ b(π), and j ∼π k, then k ∈ b(π) (i.e., the interval b(π) is isolated in π: π ≤ {b(π), b(π)c}); and

• if π, σ ∈ BNC(χ) satisfy π ∨ m(b(π)) = σ ∨ m(b(π)) then b(π) = b(σ) (i.e., any partition obtained from π by only modifying the part of π in b(π) is mapped to the same χ-interval by b).

For example, one could take b(π) to be the χ-minimal element of J which is isolated in π. Any partition π ∈ BNC(χ) with π ≤ ι must leave one element of J isolated; indeed, if one takes π ∨ J ≤ ι (the element of BNC(χ) obtained from π by joining all points lying in the same ι-monochromatic χ-intervals), it must contain a χ-interval (as any non-crossing

−1 partition, in particular sχ · (π ∨ J ), must contain an interval) and since J ≤ π ∨ J ≤ ι, any interval it contains must be isolated and maximal ι-monochromatic. Then this same interval is isolated in π.

Denote S(B) = {π ∈ BNC(χ): B ∈ π} the set of χ-non-crossing partitions in which B is a block. Note that if σ ∈ S(B) and ρ ∈ S(Bc), then σ ∧ ρ is a partition with blocks under B corresponding to ρ and blocks outside of B corresponding to σ. Further, any partition π ∈ BNC(χ) with π ≤ {B,Bc} may be expressed in this form: π ∨ m(B) ∈ S(B), π ∨ m(Bc) ∈ S(Bc), and (π ∨ m(B)) ∧ (π ∨ m(Bc)) = π.

45 Now, let n ∈ N, χ :[n] → {`, r}, and ι :[n] → I, and take z1, . . . , zn as in the deﬁnition of vaccine. Using the moment-cumulant formula and the vanishing of mixed cumulants from bi-freeness, we have

X ϕ(z1 ··· zn) = κπ(z1, . . . , zn) π∈BNC(χ) π≤ι X X = κπ(z1, . . . , zn) B∈J π∈b−1(B) X X X = κσ∧ρ(z1, . . . , zn) B∈J σ∈S(B) ρ∈S(Bc) b(σ)=B   X X X =  κρ(z1, . . . , zn) κσ\{B}(z1, . . . , zn)

B∈J σ∈S(B) ρ∈BNC(χ|B ) b(σ)=B X X = ϕ(zB)κσ\{B}(z1, . . . , zn) B∈J σ∈S(B) b(σ)=B = 0.

Essentially, in the sum of cumulants representing ϕ(z1 ··· zn), we have grouped terms together by isolated intervals, and used the fact that when we sum over the entire lattice of bi-non- crossing partitions over one of these intervals, we recover the moment corresponding to that interval, which is zero by assumption.

We remark that a bi-free family enjoys an even stronger property: one may drop the requirement that the first and last χ-intervals be centred, provided that there are at least three maximal ι-monochromatic χ-intervals. Our proof above hinged on being able to find an isolated maximal χ-interval in any χ-non-crossing partition; but notice that not only does such an interval always exist, there is always one which is neither the first nor last interval.

(ι) (ι) Lemma 2.5.4. Let (A` , Ar ) be a family of pairs of faces in a non-commutative ι∈I probability space (A, ϕ). Then the pairs of faces are bi-free if the family has vaccine.

Proof. We will show that vaccine uniquely speciﬁes mixed moments in terms of pure ones. (ι(i)) Let n ≥ 1, χ :[n] → {`, r}, ι :[n] → I, and suppose z1, . . . , zn ∈ A with zi ∈ Aχ(i) . Denote 46 by J the set of maximal ι-monochromatic χ-intervals in [n].

For each I = {a1 < ··· < aj} ∈ J , let λI be a (complex) root of the polynomial ϕ (za1 − w) ··· (zaj − w) . Then if f :[n] → J is the unique map so that i ∈ f(i) for every i, we have ϕ (z1 − λf(1)) ··· (zn − λf(n)) = 0,

as the λ’s were chosen precisely to make the vaccine property apply. Expanding this equation

gives us an expression for ϕ(z1 ··· zn) in terms of mixed moments with at most n − 1 terms;

by recursively applying the same procedure we ﬁnd an expression for ϕ(z1 ··· zn) in terms of pure moments.

(ι) D (ι) (ι)E ∗∗ (ι) Now, for ι ∈ I let ϕ be the restriction of ϕ to A` , Ar , and µ = ι∈Iϕ their bi-free product distribution, which by Lemma 2.5.3 also has vaccine. We then ﬁnd that the same expressions for joint moments in terms of pure ones hold under µ as under ϕ, which is to (ι) (ι) say that (A` , Ar are bi-free. ι∈I (ι) (ι) Theorem 2.5.5. Let (A` , Ar ) be a family of pairs of faces in a non-commutative ι∈I probability space (A, ϕ). Then the family has vaccine if and only if the pairs of faces are bi-free.

2.5.2 Conditional bi-freeness.

We remark that this condition may be extended to other settings in bi-free probability. Conditional bi-freeness was studied by Gu and Skoufranis in [11], building oﬀ of conditional free independence which was introduced by Bo˙zejko, Leinert, and Speicher [6]. We will show that conditional bi-free independence also admits a characterization in terms of χ-intervals. First, though, we take the time to introduce some notation.

Suppose that π ∈ BNC(χ). A a block B ∈ π is said to be inner if there is another block

C ∈ π and j, k ∈ C so that for every i ∈ B, j ≺χ i ≺χ k; a block which is not inner is said to be outer. This corresponds with our earlier use of the term “outer” in Subsection 2.1.2.

Let (A, ϕ) be a non-commutative probability space, and θ a state on A. The conditional

n cumulants with respect to (θ, ϕ) are multi-linear functionals Kχ : A → C deﬁned by the 47 requirement that for any z1, . . . , zn ∈ A,     X Y Y θ(z ··· z ) =  κ ((z , . . . , z )| )  K ((z , . . . , z )| ) . 1 n  χ|V 1 n V   χ|V 1 n V  π∈BNC(χ) V ∈π V ∈π V inner V outer Here κ represents the usual bi-free cumulants taken with respect to ϕ. For π ∈ BNC(χ), we

will denote by Kπ(z1, . . . , zn) the term in the above sum corresponding to π, a product of κχ|V (ι) (ι) terms and Kχ|V terms. We will say a family A` , Ar is conditionally bi-free in (A, θ, ϕ) ι∈I if it is bi-free with respect to ϕ and all mixed conditional cumulants vanish; it was shown in [11] that this is equivalent to their deﬁnition in terms of free product representations, and moreover, that being conditionally bi-free uniquely speciﬁes the mixed θ-moments in terms of the pure θ-moments and ϕ-moments.

Theorem 2.5.6. Let (A, ϕ) be a non-commutative probability space and θ : A → C a (ι) (ι) state on A. Suppose A` , Ar is a family of pairs of faces in A. Then the family is ι∈I conditionally bi-free if and only if whenever:

• n ≥ 1, χ :[n] → {`, r}, ι :[n] → I,

•J is the set of maximal ι-monochromatic χ-intervals, and

• z1, . . . , zn ∈ A are such that:

(ι(i)) – zi ∈ Aχ(i) ; and

– ϕ(zJ ) = 0 for each J ∈ J it follows that Y ϕ(z1 ··· zn) = 0 and θ(a1 ··· an) = θ(zJ ). J∈J

Proof. By the same argument as in the proof of Lemma 2.5.4 it follows that the conditions assumed above suﬃce to uniquely specify all mixed ϕ- and θ-moments in terms of pure ϕ- and θ-moments; hence if we can show that conditionally bi-free families satisfy this condition the proof will be complete. The condition on ϕ is precisely vaccine, so we need only show that our expression for mixed θ-moments is correct. 48 We take an approach similar to that of Lemma 2.5.3 for deducing the value of θ. We claim that the only terms which contribute to the value of θ in the cumulant expansion are those corresponding to partitions π < J , where once again J is the set of maximal ι-monochromatic χ-intervals. Towards this end, let b : {π ∈ BNC(χ): π ≤ ι} → J be as in

Lemma 2.5.3, with the additional constraint that b picks interior intervals whenever π J . That is, b should have the following properties:

• if π ∈ BNC(χ), j ∈ b(π), and j ∼π k, then k ∈ b(π) (i.e., the interval b(π) is isolated in π: π ≤ {b(π), b(π)c})

• if π, σ ∈ BNC(χ) satisfy π ∨ m(b(π)) = σ ∨ m(b(π)) then b(π) = b(σ) (i.e., any partition obtained from π by only modifying the part of π in b(π) is mapped to the same χ-interval by b); and

• if π ∈ BNC(χ) and π J , then b(π) is an inner block in π ∨ J .

Such functions exist: for example, one could take b(π) to be the χ-minimal element of J which is inner and isolated in π, if such exists, and the χ-minimal element of J otherwise.

Note that if π J , π must connect two intervals in J and so there must be an inner block in J ∨ π between these two intervals. As before, let S(B) = {π ∈ BNC(χ): B ∈ π}, and set Si(B) = {π ∈ S(B): B inner in π ∨ J }. We now compute much as in the proof of Lemma 2.5.3. Supposing n ∈ N, χ :[n] → {`, r}, and ι :[n] → I, and with z1, . . . , zn

49 meeting the hypotheses of the lemma:

X θ(z1 ··· zn) = Kπ(z1, . . . , zn) π∈BNC(χ) π≤ι   X  X X  =  Kπ(z1, . . . , zn) + Kπ(z1, . . . , zn)   B∈J π∈b−1(B) π∈b−1(B) B inner in π∨J B outer in π∨J   X  X X  =  ϕ(zB)Kπ\{B}(z1, . . . , zn) + Kπ(z1, . . . , zn)  −1 −1  B∈J π∈Si(B)∩b (B) π∈b (B) B outer in π∨J X = Kπ(z1, . . . , zn) π∈b−1(B) B outer in π∨J X = Kπ(z1, . . . , zn) π∈BNC(χ) π≤J Y X = KπJ ((z1, . . . , zn)|J )

J∈J πJ ∈BNC(χ|J ) Y = θ(zJ ). J∈J

Here in the last few lines we have noted that summing over all partitions sitting under J is the same as summing over partitions sitting under each interval individually, and then taking the product; this is valid since every term in Kπ is a product of terms corresponding to blocks, and each block must be contained in a single interval in J .

2.6 Bi-free multiplicative convolution.

Much as in the free setting, we can view use bi-free probability to describe a multiplicative convolution on laws.

D (ι) (ι)E Deﬁnition 2.6.1. If µι : C X` ,Xr → C are states for ι ∈ {1, 2}, there is a a unique state D (1) (2) (1) (2)E (ι) (ι) µ : C X` ,X` ,Xr ,Xr → C so that µ restricted to the algebra generated by X` ,Xr (ι) (ι) is equal to µι and the pairs (X` ,Xr ) are bi-free. This allows us to deﬁne µ1 K µ2 :

50 C hX`,Xri → C via

(1) (2) (2) (1) µ1 K µ2 (f(X`,Xr)) = µ f(X` X` ,Xr Xr ) .

There is of course a second option for deﬁning the multiplicative convolution, which would (1) (2) be to sue Xr Xr in the second coordinate; the corresponding operation is denoted by and has been studied by Voiculescu in [32] and Skoufranis in [19]. As we will see, though, the operation K ﬁts more naturally with the combinatorics of bi-free independence.

2.6.1 The Kreweras complement.

To describe the law µ1 K µ2, we will adapt the Kreweras complement approach of Nica and Speicher in [17] to our setting.

Deﬁnition 2.6.2. Let π ∈ BN C(χ). Then the Kreweras complement of π is the χ-non- crossing partition KBNC(π) deﬁned by

−1 KBNC(π) := sχ · KNC(sχ · π).

¯ ¯ Equivalently, if KBNC(π) is the maximum partition on {1¯,..., n¯} such π ∪ KBNC(π) is bi- non-crossing on the set {1, . . . , n, 1¯,..., n¯} with choice of side given by χ0(j) = χ(j) =

0 χ (¯j) and ordering such that j < ¯j on the left and ¯j < j on the right, then KBNC(π) := ¯ B ⊂ [n]: {¯j : j ∈ B} ∈ KBNC(π) .

¯ Remark 2.6.3. We point out that KBNC(π) is the unique partition on {1¯,..., n¯} such that ¯ 0 0 π ∪ KBNC(π) is χ -non-crossing and, if β = {{j, ¯j} : j ∈ [n]} ∈ BN C(χ ), then we have ¯ ¯ π ∪ KBNC(π) ∨ β = 1χ0 . Indeed, the fact that KBNC(π) is maximum ensures that any larger partition is not χ0-non-crossing, while any smaller partition must fail to connect two ¯ nodes which are connected by KBNC(π), which will then remain disconnected in ρ ∪ π ∨ β.

Example 2.6.4. Suppose that χ is given by the sequence (`, r, `, `, r, `, r, r) and

π = {{1} , {2, 4, 8} , {3} , {5, 7} , {6}} ∈ BN C(χ).

51 Then KBNC(π) = {{1, 2, 3} , {4, 6} , {5, 8} , {7}} ∈ BN C(χ), as illustrated below.

1 1¯ ¯ 22 3 3¯ 4 4¯ ¯ 55 6 6¯ ¯ 77 ¯ 88

We notice that since KNC is order reversing while sχ is order-preserving on P(n), KBNC ∼ is order-reversing. Then for π ∈ BN C(χ), we have [π, 1χ] = [KBNC(1χ),KBNC(π)] =

[0χ,KBNC(π)]. Then if f, g ∈ IA(BNC) are multiplicative,

X (f ? g)(0χ, 1χ) = f(0χ, π)g(0χ,KBNC(π)) = (g ? f)(0χ, 1χ); π∈BN C(χ)

hence f ? g = g ? f.

2.6.2 Cumulants of products.

Let n ∈ N, and suppose 0 = k0 < k1 < ··· < km = n. For χ :[m] → {`, r}, deﬁne

χˆ :[n] → {`, r} to be constant on intervals (kj−1, kj] withχ ˆ(kj) = χ(j). There is an

embedding of BNC(χ) ,→ BN C(ˆχ) via π 7→ πˆ, the unique partition with kj ∼πˆ ki if and

only if j ∼π i and {(kj−1, kj] : 1 ≤ j ≤ m} ≤ πˆ; note that this partition of intervals is

precisely 0ˆχ, while 1ˆχ = 1χˆ. Further,

{πˆ : π ∈ BN C(χ)} = 0ˆχ, 1ˆχ = 0ˆχ, 1χˆ ⊆ BN C(ˆχ)

is an interval of BNC(ˆχ) and π 7→ πˆ is a lattice morphism, so µBNC(σ, π) = µBNC(ˆσ, πˆ) for σ, π ∈ BN C(χ).

We will now exploit a useful combinatorial fact (see, e.g., [18, Proposition 10.11]): if

functions f, g : BNC(ˆχ) → C (or any ﬁnite lattice) are related via X f(π) = g(σ), σ∈BN C(χ) σ≤π 52 then X X f(τ)µBNC(τ, π) = g(ω). τ∈BN C(ˆχ) ω∈BN C(ˆχ) σ≤τ≤π ω∨σ=π

Proposition 2.6.5. Let (A, ϕ) be a non-commutative probability space, n ∈ N, 0 = k0 <

k1 < ··· < km = n, and χ :[m] → {`, r}. If π ∈ BN C(χ) and Tj ∈ Aχˆ(j) for j ∈ [n], then X κπ T1 ··· Tk1 ,Tk1+1 ··· Tk2 ,...,Tkm−1+1 ··· Tkm = κσ(T1,...,Tn). σ∈BN C(ˆχ) σ∨0ˆχ=ˆπ

Proof. Expressing the cumulant above in terms of moments via the moment-cumulant relation mentioned in Remark 2.3.7, we ﬁnd

κπ T1 ··· Tk1 ,Tk1+1 ··· Tk2 ,...,Tkm−1+1 ··· Tkm X = ϕτ T1 ··· Tk1 ,Tk1+1 ··· Tk2 ,...,Tkm−1+1 ··· Tkm µBNC(τ, π) τ∈BN C(χ) τ≤π X = ϕτˆ(T1,...,Tn)µBNC(ˆτ, πˆ) τ∈BN C(χ) τ≤π X = ϕσ(T1,...,Tn)µBNC(σ, πˆ) σ∈BN C(ˆχ) 0ˆχ≤σ≤πˆ X = κσ(T1,...,Tn). σ∈BN C(ˆχ) σ∨0ˆχ=ˆπ

2.6.3 Multiplicative convolution.

Theorem 2.6.6. Let (A, ϕ) be a non-commutative probability space, and suppose that the (ι) (ι) (1) (2) (2) (1) pairs of faces (zi )i∈I , (zj )j∈J are bi-free; set zi = zi zi for i ∈ I, and zj = zj zj ι∈{1,2} for j ∈ J. Then for every α :[n] → I ` J, we have

X (1) (1) (2) (2) κχα (zα(1), . . . , zα(n)) = κπ zα(1), . . . , zα(n) κKBNC(π) zα(1), . . . , zα(n) . π∈BN C(χα)

Our proof here is inspired by the presentation of the free version found in [18, Theorem 14.4]. 53 ` Proof. Deﬁne αb : [2n] → I J by αb(2k − 1) = αb(2k) = α(k) for k ∈ [n], and deﬁne ι : [2n] → {1, 2} by    1 if α(k) ∈ I  2 if α(k) ∈ I ι(2k − 1) = and ι(2k) = .  2 if α(k) ∈ J  1 if α(k) ∈ J

Using Proposition 2.6.5 we see that

X ι(1) ι(2) ι(2n−1) ι(2n) κχ(zα(1), . . . , zα(n)) = κπ zα(1), zα(1), . . . , zα(n) , zα(n) π∈BN C(χ ) αb π∨σ=1χ αb where σ = {(1, 2), (3, 4),..., (2n − 1, 2n)}. By bi-freeness and Theorem 2.4.7, the mixed cumulants above vanish, and the only partitions which survive correspond to pairs π(1), π(2) such that π(i) is a bi-non-crossing partition on the nodes in ι−1(i), such that the pair taken

(1) (2) together remain bi-non-crossing on [2n], and π ∪ π ∨ σ = 1χαˆ . But as we noted in Remark 2.6.3, this is exactly the condition that π(2) is the Kreweras complement of π(1) (on relabelled indices). Thus the claimed equation follows.

If we restrict the above theorem to the case that each of I and J contains a single element, we have a formula for how cumulants behave under bi-free multiplicative convolu-

tion. By summing over KBNC(π) instead of π, we also ﬁnd that multiplicative convolution is commutative:

µ1 K µ2 = µ2 K µ1.

2.7 An operator model for pairs of faces.

In this section, we will construct an operator model for a two-faced family in a non- commutative probability space. The model will generalize the operator model of Nica, [15], from the free setting to the bi-free setting. Our goal is to recognize the variables as formal sums of (densely-deﬁned unbounded) operators on a Fock space, for which the pairings of

the form hei1 ⊗ · · · ⊗ ein , T ej1 ⊗ · · · ⊗ ejm i make sense and are ﬁnite for any T in the algebra generated by the representatives of the variables.

54 2.7.1 Nica’s operator model.

We begin by reviewing Nica’s operator model, with the intent of making the description of our

model more understandable. Suppose that we are given a law µ : C hX1,...,Xni → C, and n let κ be the corresponding cumulant functional. Let H := F(C ), and take {e1, . . . , en} to n ∗ ∗ be the standard orthonormal basis of C , and let us use the shorthand `j = `(ej), `j = ` (ej) for the corresponding creation and annihilation operators. We extend the vacuum state on H to the space on which we are working by ω(T ) = hΩ,T Ωi.

Formally, we deﬁne:

X X Θµ := I + κk(Xi1 ,...,Xik )`ik ··· `i1

k≥1 i1,...,ik∈[n]

∗ ∗ X X Zi := ì Θµ = ì + κk+1(Xi1 ,...,Xik ,Xi)ìk ··· ì1 ,

k≥0 i1,...,ik∈[n] where we understand an empty product of creation operators as the identity operator.

Proposition 2.7.1. The variables X1,...,Xn with respect to ϕ and Z1,...,Zn with respect to ω have the same law.

Proof. Fix i1, . . . , ik ∈ [n]. On the one hand, from the moment cumulant formula, we have

X ϕ(Xi1 ··· Xik ) = κ(Xi1 ,...,Xik ). π∈N C(k)

On the other hand, let us consider the expansion of Zi1 ··· Zik Ω. Note that in order for a term to survive when paired with Ω, each creation operator must be paired with an annihilation operator to its left, and each annihilation operator must be paired with a creation operator to its right; the indices of these operators must agree. Then we can associate to each surviving term a unique element of NC(k) by taking the minimal partition which

satisﬁes that a ∼ b whenever Zia introduces a creation operator paired with the annihilation operator from Zib . This non-crossing partition is unique, and for each non-crossing partition we may ﬁnd such a term which survives. But now the weight of the term corresponding to π is precisely the product over the blocks of B of the corresponding cumulants, namely,

55 κπ(Xi1 ,...,Xik ). Thus

X ω(Zi1 ··· Xik ) = κ(Xi1 ,...,Xik ). π∈N C(k)

We think of the operator as follows: each term in the sum of Θµ corresponds to all possible blocks which may occur in a non-crossing partition. In the product `∗ Θ `∗ Θ ··· `∗ Θ , each i1 µ i2 µ ik µ annihilation operator “ﬁlls” a slot in a block which has been introduced by a prior Θµ, while each Θµ may introduce a new block at the current point in the progress of building the partition. We introduce many potential blocks by Θµ, but those which are not valid for the product we are currently building cannot be completed by the appropriate annihilation operators and so do not contribute.

2.7.2 Skeletons corresponding to bi-non-crossing partitions.

We now aim to ﬁnd an appropriate extension of this model to the bi-free setting. One may be tempted to try to model left variables by left creation and annihilation operators, and right variables by right creation and annihilation operators. Unfortunately the fact that in general left and right variables from the same pair of faces need not commute with each other means that such a model cannot possibly work; we need more complicated models to capture this behaviour.

Deﬁnition 2.7.2. Let χ :[n] → {`, r} and let π ∈ BN C(χ). A skeleton on π is the data ` (α, k) where α :[n] → I J is such that χα = χ, and 0 ≤ k ≤ n. We will represent this data by drawing the diagram corresponding to π, labelling the nodes according to α, and ﬁlling the bottom k nodes while leaving the top n−k empty. If k = n, the skeleton is said to be completed; if k = 0, it is said to be empty; and if k = 1 it is said to be a starter skeleton.

Example 2.7.3. Suppose χ corresponds to the sequence (`, `, r, `, r), ﬁx α : [5] → I ` J,

56 and suppose π = {{1, 3} , {2, 4, 5}}. Then the six skeletons corresponding to π and α are:

α(1) α(1) α(1) α(2) α(2) α(2) α(3) α(3) α(3) α(4) α(4) α(4) α(5) α(5) α(5)

We will use skeletons to make it easier to keep track of what operators have acted so far, and how new ones could potentially be added. Skeletons can be related to Nica’s model

as well: one thinks of any term `i1 ··· `ik in the sum in Θµ as corresponding to adding to introducing an empty skeleton consisting of a single block to the current skeleton, with nodes

∗ labelled by i1, . . . , ik, while each annihilation operator `j ﬁlls in the next empty node if it is labelled by j, and otherwise returns 0. A skeleton corresponds to a tensor product of the vectors with labels corresponding to its unﬁlled nodes; the partition corresponding to the skeleton contributes only to the scalar portion of the vector. At the end, only completed skeletons will contribute as only they will correspond to vectors in CΩ.

Example 2.7.4. Let us consider the product Z1Z2Z3Z4 in Nica’s model, and examine how the term corresponding to κ2(X1,X4)κ2(X2,X3) may be realized. This term corresponds to

∗ ∗ ∗ ∗ choosing κ2(X1,X4)`4`4`1 from Z4, κ2(X2,X3)`3`3`2 from Z3, `2 from Z2, and `1 from Z1. 1 1 1 1 2 2 2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ `1`2`3`3`2`4`4`1Ω = `1`2`3`3`2 = `1`2 3 = `1 3 = 3 4 4 4 4

2.7.3 A construction.

We will now construct our operator model for pairs of faces, motivated by our realization of Nica’s operator model. Above, the model constructed all weighted non-crossing partitions by using creation operators to glue in full non-crossing blocks and annihilation operators to 57 approve or reject non-crossing diagrams. As the combinatorics of pairs of faces is dictated by bi-non-crossing partitions, we must construct the appropriate creation operators to glue together bi-non-crossing partitions. However, unlike with non-crossing partitions where there is only one way to glue in a full block at any given point, there may be multiple or no ways to glue one bi-non-crossing skeleton into another. As such, the description of the appropriate creation operators is more complicated.

Let z = ((zi)i∈I , (zj)j∈J ) be a two-faced family in (A, ϕ). We will construct our model as formal sums of products of creation and annihilation operators on the Fock space H :=

I ` J ` F C , with {ek : k ∈ I J} an orthonormal basis. ` For α :[n] → I J, we will deﬁne (unbounded) operators Tα ∈ L(H) which will play the roles of the terms in the sum in the deﬁnition of Θµ in Nica’s model; that is, each will add an appropriate empty skeleton. We aim to construct an operator Θ of the form

X X Θ := I + κχα (zα(1), . . . , zα(n))Tα, n≥1 α:[n]→I ` J in such a way that the variables

∗ ∗ X X Zk := `kΘ = `k + κα(zα(1), . . . , zα(n), zk)Tα n≥0 α:[n+1]→I ` J α(n+1)=k give us the correct distribution.

Though we will often speak of actions on skeletons, one can recover the context of H by letting a partially completed skeleton correspond to the vector formed by taking the tensor product of the basis elements matching the labels of its open nodes, from bottom to top, and weighting it based on which cumulants have been chosen. For example, the skeleton

α(1) α(2) α(3) α(4) α(5)

corresponds to the vector eα(3) ⊗eα(2) ⊗eα(1) and will be weighted by the product of cumulants

κ(zα(2), zα(5))κ(zα(1), zα(4))κ(zα(3)). The key point here is that the only choices of future 58 Zk which yield a non-zero Ω component when applied to such a vector have annihilation operators in the correct order. In the above example, in order for this skeleton to make a

contribution to the final term, we must act on it by Zα(3), Zα(2), and Zα(1) in that order (though other variables may occur between them). Since the closed nodes of the skeleton only effect the resulting quantity in terms of its weight and cannot affect the action of future operators (as indeed they must not, for the vector has forgotten them) we will sometimes truncate diagrams of skeletons to show only the open nodes. It is implied that there may be significantly more nodes and blocks below the bottom of the diagrams that follow, but their

representation is eschewed. Likewise, in order to ensure that Tα is well-deﬁned, we cannot have behaviour depending on which partial skeletons have been chosen, but only the choice of side and of labels of the open nodes.

For n = 1, we deﬁne Tα := `α(1). In this setting, one may think of Tα as adding an empty skeleton in the lowest possible position with a single open node on the left or on the right depending on whether α(1) is in I or J. For example,

β(1) β(1) β(1) β(2) β(2) β(2) Tα α(1) or α(1) , β(3) β(3) β(3) β(4) β(4) β(4)

depending on whether α(1) is in I or J. Observe that Tα adds an open node in the lowest valid location (i.e., immediately above all closed nodes); this behaviour will be mimicked by the other Tα as well. That is, the lowest open node added will always be added directly above the highest closed node.

Let Σ : H ⊕ H → H be deﬁned by

X Σ(f1 ⊗ · · · ⊗ fn, fn+1 ⊗ · · · ⊗ fn+m) := fσ(1) ⊗ · · · ⊗ fσ(n+m), σ where the sum is over all permutations σ ∈ Sn+m so that σ|[1,n] and σ|[n+1,n+m] are increasing; that is, σ interleaves the sets [n] and {n + 1, . . . , n + m}. Note that Σ(ξ, Ω) = ξ = Σ(Ω, ξ).

59 As an example,

Σ(e1 ⊗ e2, e3 ⊗ e4) = e1 ⊗ e2 ⊗ e3 ⊗ e4 + e1 ⊗ e3 ⊗ e2 ⊗ e4 + e3 ⊗ e1 ⊗ e2 ⊗ e4

+ e1 ⊗ e3 ⊗ e4 ⊗ e2 + e3 ⊗ e1 ⊗ e4 ⊗ e2 + e3 ⊗ e4 ⊗ e1 ⊗ e2.

We will use Σ to account for the fact that nodes on the right may be added with any order to nodes on the left to obtain a valid skeleton.

For α :[n] → I ` J we deﬁne

Tα(Ω) := `α(n)`α(n−1) ··· `α(1)(Ω) = eα(1) ⊗ · · · ⊗ eα(n).

Note that this corresponds to taking a completed skeleton (possibly with no nodes), and adding the empty skeleton corresponding to α above it.

We will now deﬁne Tα for n ≥ 2 on tensors of basis elements, and extend by linearity to their span (which, together with Ω, is dense in H). We consider only the case α(n) ∈ I,

as the case when α(n) ∈ J will be similar. Let η = eβ(m) ⊗ · · · ⊗ eβ(1) ∈ H, where β : {1, . . . , m} → I ` J.

If β−1(I) = ∅, let k = max ({0} ∪ α−1(J)), so that k is the lowest of the nodes to be added by which falls on the right. We deﬁne Tα(η) as follows:

Tα(η) := eα(n) ⊗ Σ eα(n−1) ⊗ · · · ⊗ eα(k+1), eβ(m) ⊗ · · · ⊗ eβ(1) ⊗ eα(k) ⊗ · · · ⊗ eα(1).

This is mimicking the action of adding a new skeleton to the existing skeleton. In order to ensure that no crossings are introduced, all new nodes on the right must be placed above all existing nodes on the right; before any new right nodes are added, though, nodes on the left can be added freely. One should think of this as the sum of all valid partially completed skeletons where the old skeleton is below and to the right of starter skeleton corresponding to α, with the node corresponding to α(n) in the lowest possible position.

Example 2.7.5. If α : [4] → I ` J satisﬁes α−1(I) = {1, 3, 4} and α−1(J) = {2}, and j ∈ J, then

Tα(ej) = eα(4) ⊗ eα(3) ⊗ ej ⊗ eα(2) ⊗ eα(1) + eα(4) ⊗ ej ⊗ eα(3) ⊗ eα(2) ⊗ eα(1). 60 This action corresponds to the following diagram:

α(1) α(1) α(2) α(2) α(3) Tα j j + j α(3) α(4) α(4)

The purpose of allowing multiple diagrams is that the cumulant corresponding to a bi-non- crossing diagram for a sequence of operators is equal to the same cumulant for the sequence of operators obtained by interchanging the k-th and (k + 1)-th operators and the k-th and (k + 1)-th nodes in the bi-non-crossing diagram provided k and k + 1 are in diﬀerent blocks and on diﬀerent sides of the diagram. In the end, a sequence of annihilation operators can complete at most one skeleton and will produce the correct completed skeleton for a given sequence of operators.

Note that the node labelled j above must have been introduced by some earlier operator, and so must be connected to something below the diagram. It is not possible that j is isolated; in bi-non-crossing partitions where j is isolated, it will be added to the diagram by a later operator.

As a further example, suppose α : [3] → I and j1, j2 ∈ J. Then

Tα(ej2 ⊗ ej1 ) = eα(3) ⊗ eα(2) ⊗ eα(1) ⊗ ej2 ⊗ ej1 + eα(3) ⊗ eα(2) ⊗ ej2 ⊗ eα(1) ⊗ ej1

+ eα(3) ⊗ eα(2) ⊗ ej2 ⊗ ej1 ⊗ eα(1) + eα(3) ⊗ ej2 ⊗ eα(2) ⊗ eα(1) ⊗ ej1

+ eα(3) ⊗ ej2 ⊗ eα(2) ⊗ ej1 ⊗ eα(1) + eα(3) ⊗ ej2 ⊗ ej1 ⊗ eα(2) ⊗ eα(1)

61 This action corresponds to the following diagram:

j1 j1 α(1) j2 α(1) j1 α(1) j2 j2 j Tα 1 α(2) + α(2) + α(2) j 2 α(3) α(3) α(3)

j1 α(1) α(1) α(1) j1 α(2) α(2) α(2) j1 + j2 + j2 + j2 α(3) α(3) α(3)

Note that we could equally well have drawn a diagram where j1 and j2 were not connected; however, these two diagrams correspond to proportional vectors and the diﬀerence between them is only on the scalar level.

Now, suppose that β−1(I) 6= ∅, and let k = max (β−1(I)). This corresponds to a partially completed skeleton with open nodes on both the left and right, where the lowest open node

th on the left is the k from the top. We set Tα(η) = 0 if α(t) ∈ J for some t, since the partially completed skeleton has open nodes on the left and right we cannot add the empty skeleton of α without introducing a crossing, since the lowest node of α is on the left. Otherwise α(t) ∈ I for all t ∈ {1, . . . , n}, and we set

Tα(η) := eα(n) ⊗ Σ eα(n−1) ⊗ · · · ⊗ eα(1), eβ(m) ⊗ · · · ⊗ eβ(k+1) ⊗ eβ(k) ⊗ · · · ⊗ eβ(1).

One can think of this as the sum of all valid partially completed skeletons where the empty skeleton of α sits below the lowest open node on the left of the old skeleton.

Example 2.7.6. If α : [2] → I ` J has α(2) ∈ I, α(1) ∈ J and i ∈ I, then

Tα(ei) = 0.

This is because there is no way to glue the empty skeleton corresponding to α into the partially completed skeleton without introducing a crossing while placing the lowest node of 62 α at the bottom of the diagram (directly above the highest closed node): i α(1) α(2) .

If α : {1, 2} → I and i ∈ I, j, j0 ∈ J then

Tα (ej ⊗ ei ⊗ ej0 ) = eα(2) ⊗ eα(1) ⊗ ej ⊗ ei ⊗ ej0 + eα(2) ⊗ ej ⊗ eα(1) ⊗ ei ⊗ ej0 .

This action corresponds to the following diagram: j0 j0 j0 i i i Tα j j + α(1) α(1) j α(2) α(2) .

As Tα has been deﬁned on an orthonormal basis, we may extend by linearity to obtain a densely deﬁned operator on H; note that Tα may not be bounded due to the action of Σ.

On the other hand, if α :[n] → I then Tα acts on the Fock subspace generated by {ei}i∈I as

`α(n) ··· `α(1). Thus, if one considers only left variables, the resulting operators are precisely those of Nica’s model.

We deﬁne Tα in a similar manner when α(n) ∈ J.

2.7.4 The operator model for pairs of faces.

With the above construction, the operator model for a pair of faces is at hand. Theorem 2.7.7. Let z = (zi)i∈I , (zj)j∈J be a pair of faces in a non-commutative probability space (A, ϕ). With notation as in Subsection 2.7.3, consider the formal sum X X Θz := I + κα(z)Tα, n≥1 α:[n]→I ` J ` ∗ and for k ∈ I J, set Zk := `kΘz. If T ∈ alg({Zk}k∈I ` J ) then hΩ,T Ωi is well-deﬁned. Moreover, with respect to the vacuum state ω(T ) = hΩ,T Ωi, the joint distribution of the family {Zk}k∈I ` J is the same as the joint distribution of z with respect to ϕ. 63 Before we begin the proof, we give the following example.

Example 2.7.8. In this example, let I = {1} and J = {2}. We will examine how the completed skeleton below is constructed for Z1Z2Z1Z1Z2Z2Z1Z2Z1Z1.

1 2 1 1 2 2 1 2 1 1

∗ First κ(21)(z)`1T(21) is applied to Ω to get the partially completed skeleton

2 1 .

∗ Then κ(1211)(z)`1T(1211) is applied to obtain the following collection of partially completed skeletons: 1 1 2 2 2 1 1 2 1 1 1 1 .

∗ Applying κ(22)`2T(22) then gives the following collection of partially completed skeletons (where the first two below are from the first above and the third below is from the second above): 1 1 1 2 2 2 2 2 1 2 1 2 1 2 2 2 2 2 1 1 1 1 1 1 . 64 ∗ Now applying κ(11)`1T(11) gives the following collection of partially completed skeletons (where the first below is from the first above, the second and third below are from the second above, and the last three are from the third above):

1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 1 1 1 2 1 1 2 2 1 1 2 1 2 1 2 1 1 2 1 2 2 1 1 1 1 1 1 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 .

∗ Applying `2 then gives the following collection of partially completed skeletons (where the ﬁrst, second, and fourth diagrams above were destroyed):

1 1 1 2 2 2 2 1 1 1 2 1 1 1 2 2 2 2 1 1 1 2 2 2 1 1 1 1 1 1 .

∗ Next, applying `2 removes all but the last diagram to give:

1 2 1 1 2 2 1 2 1 1 . 65 ∗ ∗ ∗ ∗ Applying `1`2`1`1 then gives us the desired diagram. We also see the diagram was weighted by

κ(21)(z)κ(1211)(z)κ(22)κ(11)(z) which is the correct product of bi-free cumulants for this bi-non-crossing partition.

Proof of Theorem 2.7.7. Let α :[n] → I ` J. To see that

ω(Zα(1) ··· Zα(n)) = ϕ(zα(1) ··· zα(n)), we must demonstrate that the sum of over all

∗ ∗ a Ak ∈ `α(k) ∪ κβ(z)`α(k)Tβ | β : {1, . . . , m → I J} of

hΩ,A1 ··· AnΩi

∗ is precisely ϕ(zα(1) ··· zα(n)). (Note that `α(k)Tβ = 0 unless β(m) = α(k).) This suﬃces as these are precisely the terms that appear in expanding the product Zα(1) ··· Zα(n). By construction A1 ··· An acting on Ω corresponds to creating a (sequence of) partially completed skeletons and hΩ,A1 ··· AnΩi will be the weight of the skeleton if the skeleton is complete and otherwise will be zero. Since

X ϕ(zα(1) ··· zα(n)) = κπ(z).

π∈BN C(χα) it suﬃces to show that there is a bijection between completed skeletons and elements π of

BNC(χα), and that the weight of the skeleton is the corresponding cumulant.

Observe that after Ak is applied, the bottom n − k + 1 nodes of the partially completed skeleton will be closed, as Ak itself either closed an open node which was already present or added a new starter skeleton containing one closed node and zero or more open nodes. In particular, the (n − k + 1)-th node from the bottom must be on the side corresponding

∗ to α(k) since it was closed by `α(k). Thus when we have applied A1 ··· An, any completed skeleton surviving has precisely n nodes and structure arising from α.

66 From a bi-non-crossing partition π ∈ BNC(α), we can recover the choice of A1,...,An which produces it. To do so, for each block V = {k < . . . < k }, we let A = `∗ for i 6= t, 1 t ki ki ∗ and with βV (i) = α(ki), we set Akt = κβV (z)`kt TβV . Indeed, the partially created skeletons

created by Ak ··· An agree with π on the bottom n − k + 1 nodes. Moreover, given any

0 0 other product A1 ··· An which diﬀers from A1 ··· An, consider the greatest index k so that 0 0 0 Ak 6= Ak. Then all partially completed skeletons in Ak ··· An and Ak ··· An agree in structure for their bottom n − k nodes, while the next either starts a new block in one case but not the other or starts new blocks of diﬀerent shapes; thus there is only one sequence of choices of Aj for each bi-non-crossing partition. Finally, note that if βV corresponds to the block

V ∈ π as above, then κβV (z) = κπ|V (z) and so the total weight on the skeleton is precisely

κπ(z).

Remark 2.7.9. In Theorem 7.4 of [30], an operator model for the bi-free central limit distributions was given as sums of creation and annihilation operators on a Fock space as

∗ ` follows. Let C = (Ck,l)k,l∈I ` J be a matrix with complex entries, and let h, h : I J → H

∗ be maps into a Hilbert space H so that Ck,l = hh (k), h(l)i. Then if we deﬁne

∗ ∗ ∗ ∗ zi = `(h(i)) + ` (h (i)) for i ∈ I and zj = r(h(j)) + r (h (j)) for j ∈ J,

we ﬁnd that zi and zj are bi-free central limit distributions with covariance ω(zkzl) = Ck,l (that is, the only non-vanishing cumulants for z are those of second order, which are given by the matrix C).

It is interesting that the operator model from Theorem 2.7.7 uses diﬀerent operators. Indeed for i, i0 ∈ I and j ∈ J, one can check that

X X ∗ ∗ T(i,i0) = `i0 `α(1) ··· `α(n)`i`α(n) ··· `α(1) n≥0 α:[n]→J and

T(j,i0) = `i0 rjP where P is the projection onto the Fock subspace of H generated by {ej}j∈J , and rj is the ` right creation operator corresponding to ej. Therefore, if Ck1,k2 = ϕ(zk1 zk2 ) for k1, k2 ∈ I J 67 with z a bi-free central limit distribution, Theorem 2.7.7 produces the operators

∗ X ∗ Zk = `k + Ck0,k`kT(k0,k) k0∈I ` J ∗ ∗ which are very different from `k + `k (if k ∈ I) and rk + rk (if k ∈ J) proposed in [30]. ∗ ∗ The main issues with the model involving ì, ì , rj, rj , : i ∈ I, j ∈ J is that the vectors obtained by applying the algebra generated by these operators to Ω do not generate the full Fock space: indeed, they only generate vectors of the form

ei1 ⊗ · · · ⊗ ein ⊗ ejm ⊗ · · · ⊗ ej1 where n, m ≥ 0, i1, . . . , in ∈ I, and j1, . . . , jm ∈ J. It is not diﬃcult to see that the vectors ∗ ∗ obtained by the algebra generated by Li ,Lj ,T(i,i),T(j,j) : i ∈ I, j ∈ J applied to Ω generate the full Fock space.

2.8 Additional cases of bi-free independence.

We take a moment to describe some useful techniques for producing examples of families which are bi-freely independent. The techniques covered here apply in the broader context of Chapter 3, but are more easily stated in this scalar-valued setting, and so we will cover them before proceeding to that greater generality.

2.8.1 Conjugation by a Haar pair of unitaries.

Recall that a Haar unitary is a unitary u ∈ A such that ϕ(uk) = 0 for non-zero k ∈ Z.A concrete example is given by pointwise multiplication by e2πix on L2([0, 1], dλ). The following result is well-known, and motivates us to search for a similar statement in the bi-free settings.

Proposition 2.8.1. Suppose that (A, ϕ) is a non-commutative probability space, and B ⊂ A is free from a Haar unitary u ∈ A. Then ukAu−k are free, and ukAu−k is equal in k∈Z distribution to A.

We will not sketch a proof of this proposition here, though it will follow from our bi-free result at the end of this section. 68 Deﬁnition 2.8.2. Suppose (A, ϕ) is a non-commutative probability space. A Haar pair of unitaries is a pair (u`, ur) of invertible elements of A which agree in distribution with the pair (u, u∗) where u is a Haar unitary; that is, if for every word f in non-commuting

∗ ∗ ∗ ∗ indeterminates X, X , Y , and Y , we have ϕ(f(u`, u` , ur, ur)) = 0 unless

degX (f) + degY ∗ (f) = degX∗ (f) + degY (f),

∗ ∗ and in that case ϕ(f(u`, u` , ur, ur)) = 1.

∗ We remark here that although ur is distributed as u` , they are not necessarily equal.

This is necessary because we will want to take things to be bi-free from the pair (u`, ur), but any left variable bi-free from that pair must commute in distribution with ur while being

∗ free from u`. Indeed, if we were to begin with the pair (u, u ) and take its bi-free product

∗ ∗ with a pair (A`, Ar) we would ﬁnd that u is represented as λ(u) while u is as ρ(u ) on the free product space, and certainly λ(u)∗ = λ(u∗) 6= ρ(u∗).

Theorem 2.8.3. Suppose that (A, ϕ) is a non-commutative probability space, and (A`, Ar) ⊂

A is a pair of faces bi-free from a Haar pair of unitaries (u`, ur) in A. Then the family of k −k k −k pairs of faces (u A`u , u Aru ) are bi-free and identically distributed. ` ` r r k∈Z

Proof. We will ﬁrst show that each individual pair of faces has the same distribution as the original. To that end, ﬁx k ∈ Z \{0}, χ :[n] → {`, r}, and xi ∈ Aχ(i). Note that if χ is constant, we have

k −k k −k k −k ϕ uχ(1)x1uχ(1) ··· uχ(n)xnuχ(n) = ϕ uχ(1)x1 ··· xnuχ(n) .

k k Then from freeness, and using the fact that ϕ(uχ(1)) = 0 = ϕ(uχ(n)), we have that this agrees with ϕ(x1 ··· xn). (Note that we do not require ϕ to be tracial for this result, although the proof is even simpler with that additional assumption.)

Therefore let us assume χ is non-constant. Let us further assume that our representation

∗ is on a free product space with (u`, ur) the image under λ and ρ of a pair (u, u ), so in k k k −k k −k particular, ϕ(T u` ur ) = ϕ(T ) for any T ∈ A (since λ(u )ρ u ξ = λ(u u )ξ = ξ). k k ˚ k k Similarly, we have ϕ(u` rr T ) = ϕ(T ) as is T ξ = λξ + η with η ∈ X then u` ur T ξ = λξ + 69 k −k ˚ k −k k −k λ(u )ρ(u )η and the later term remains in X. Then take y = uχ(1)x1uχ(1) ··· uχ(n)xnuχ(n); we wish to show that

ϕ (y) = ϕ (x1 ··· xn) .

Notice that since u` commutes with ur and Ar, we may collect and cancel all except the ﬁrst

and last instance of u`’s; we may do the same with the ur’s. We may then move the remaining

u` and ur terms to the top and bottom of the corresponding bi-non-crossing diagram and we conclude that k k −k −k ϕ (y) = ϕ u` ur x1x2 ··· xnu` ur = ϕ(x1 ··· xn).

We now wish to show bi-free independence. Let n ≥ 1, χ :[n] → {`, r}, and k1, . . . , kn ∈

ki −ki Z. Choose z1, . . . , zn ∈ A so that zi = uχ(i)xiuχ(i) with xi ∈ Aχ(i), and suppose that

whenever {i1 < ··· < ik} is a maximal monochromatic χ-interval, ϕ(zi1 ··· zik ) = 0. By the

above, this is equivalent to saying ϕ(xi1 ··· xik ) = 0. Using the same tricks as in the ﬁrst part of the argument, we may cancel all u’s which do not occur at the beginning or end of a

χ-interval. Then vaccine together with the bi-freeness of (A`, Ar) and (u`, ur) tells us that

ϕ(z1 ··· zn) = 0.

2.8.2 Bi-free independence for bipartite pairs of faces.

(ι) (ι) h (i) (j)i A family of pairs of faces (A` , Ar ) is said to be bipartite if A` , Ar = 0 for every ι∈I i, j ∈ I. In [30, 31], Voiculescu made special focus on bipartite families of pairs of faces, as this additional assumption provides much more power for determining their behaviour. For example, to fully describe the joint distribution of such a family it suﬃces to specify the two-bands moments: the moments of a product of left variables followed by a product of right variables. We will show that in this context it can be much simpler to verify bi-free independence.

(ι) (ι) Theorem 2.8.4. Let (A` , Ar ) be a bipartite family of pairs of faces in a non- ι∈I commutative probability space (A, ϕ), acting on a vector space with speciﬁed state vector ˚ (ι) (ι) (X, X, ξ). Suppose, further, that for every ι ∈ I and every T ∈ Ar there is S ∈ A` such (ι) (ι) (ι) that T ξ = Sξ. Then (A` , Ar ) are bi-free if and only if A` are free. ι∈I ι∈I 70 (ι) (ι) Proof. As we remarked in Subsection 1.1.5 of the introduction, bi-freeness of (A` , Ar ) ι∈I implies the freeness of the left faces. It is up to us here to demonstrate the converse.

Notice that if T,S are as in the statement of the theorem, the condition T ξ = Sξ tells us that for any Q ∈ A we have ϕ(QT ) = ϕ(QS). Now, take χ :[n] → {`, r} and ι :[n] → I, (ι(i)) and let zi ∈ Aχ(i) be such that whenever {i1 < ··· < ik} is a maximal monochromatic χ-

interval, ϕ(zi1 ··· zik ) = 0. As all left variables commute with all right variables, we may assume that there is some 0 ≤ a ≤ n so that χ(i) = ` if and only if i ≤ a. For i > a,

let yi be so that ziξ = yi. Then ϕ(z1 ··· zn) = ϕ(z1 ··· zayn ··· ya+1) by repeatedly changing the rightmost z to its corresponding y, and then commuting it past the remaining right- side z’s. Notice that this also preserves the centredness of the χ-intervals we care about, as each operation of commuting or replacing a right operator by a left does not aﬀect the moment of the product, while the same operators remain in the same intervals throughout. Having done this, grouping adjacent operators which come from the same family yields us with an alternating product of centred left variables, which vanishes by ordinary freeness. (ι) (ι) Thus 0 = ϕ(z1 ··· zayn ··· ya+1) = ϕ(z1 ··· zn) and vaccine implies that (A` , Ar ) are ι∈I bi-free.

(ι) (ι) Corollary 2.8.5. Let (A` , Ar ) be a family of pairs of faces in a non-commutative ι∈I probability space (A, ϕ) acting on a vector space with speciﬁed state vector (X, X,˚ ξ). Suppose (ι) (ι) that for every ι ∈ I, A` ξ = Ar . Then the following three conditions are equivalent:

(ι) • the family A` is free; ι∈I

(ι) • the family Ar is free; and ι∈I

(ι) (ι) • the family of pairs of faces (A` , Ar ) is bi-free. ι∈I

71 CHAPTER 3

Amalgamated bi-free probability.

We now turn to an examination of the amalgamated setting of bi-free probability. Section 8 of [30] laid the framework for generalizing B-valued free probability to the bi-free setting; our goal here is to make use of the combinatorial tools we have developed in Chapter 2 to explore operator-valued bi-free probability in much greater depth.

3.1 Bi-free families with amalgamation.

In this section, we will recall and develop the structures from [30, Section 8] necessary to discuss bi-freeness with amalgamation. Throughout, B will denote a unital algebra over C.

3.1.1 Concrete structures for bi-free probability with amalgamation.

To begin the necessary constructions in the amalgamated setting, we need an analogue of a vector space with a speciﬁed vector state.

Deﬁnition 3.1.1. A B-B-bimodule with a speciﬁed B-vector state is a triple (X , X˚, p) where X is a direct sum of B-B-bimodules

X = B ⊕ X˚,

and p : X → B is the linear map p(b ⊕ η) = b.

˚ Given a B-B-bimodule with a speciﬁed B-vector state (X , X , p), for b1, b2 ∈ B and η ∈ X we have

p(b1 · η · b2) = b1p(η)b2. 72 Definition 3.1.2. Given a B-B-bimodule with a specified B-vector state (X , X˚, p), let L(X ) denote the set of linear operators on X . Given b ∈ B, we define two operators Lb,Rb ∈ L(X ) by

Lb(η) = b · η and Rb(η) = η · b for η ∈ X .

In addition, we deﬁne the unital subalgebras L`(X ) and Lr(X ) of L(X ) by

L`(X ) := {T ∈ L(X ) | TRb = RbT for all b ∈ B}

Lr(X ) := {T ∈ L(X ) | TLb = LbT for all b ∈ B} .

We call L`(X ) and Lr(X ) the left and right algebras of L(X ), respectively.

Note L`(X ) consists of all operators in L(X ) that are right B-linear; that is, if T ∈ L`(X ) then

T (η · b) = T (Rb(η)) = Rb(T (η)) = T (η) · b for all b ∈ B and η ∈ X . This may seem counter-intuitive; however, we take our left (resp. right) face to be a sub-algebra of L`(X ) (resp. Lr(X )), and we would like to think of right multiplication by B as a right variable. One sees from the B-B-bimodule structure that b 7→ Lb is a homomorphism, b 7→ Rb is an anti-homomorphism, and the ranges of these maps commute. Hence

{Lb | b ∈ B} ⊆ L`(X ) and {Rb | b ∈ B} ⊆ Lr(X ).

Thus, in the context of this paper, L`(X ) consists of ‘left’ operators and Lr(X ) consists of ‘right’ operators.

As we are interested in L(X ) and amalgamating over B, we will need an “expectation” from L(X ) to B.

Definition 3.1.3. Given a B-B-bimodule with a specified B-vector state (X , X˚, p), we define the linear map EL(X ) : L(X ) → B by

EL(X )(T ) = p(T (1B ⊕ 0))

for all T ∈ L(X ). We call EL(X ) the expectation of L(X ) onto B. 73 The following important properties justify calling EL(X ) an expectation.

Proposition 3.1.4. Let (X , X˚, p) be a B-B-bimodule with a speciﬁed B-vector state. Then

EL(X )(Lb1 Rb2 T ) = b1EL(X )(T )b2

for all b1, b2 ∈ B and T ∈ L(X ), and

EL(X )(TLb) = EL(X )(TRb) for all b ∈ B and T ∈ L(X ).

Proof. If b1, b2 ∈ B and T ∈ L(X ), we see that

EL(X )(Lb1 Rb2 T ) = p(Lb1 Rb2 T (1B ⊕ 0)) = p(Lb1 Rb2 (E(T ) ⊕ η))

= p((b1E(T )b2) ⊕ (b1 · η · b2)) = b1E(T )b2

˚ for some η ∈ X . The second result holds as Lb(1B ⊕ 0) = b = Rb(1B ⊕ 0).

To complete this section, we recall the construction of the reduced free product of B-B- bimodules with speciﬁed B-vector states. This will be similar in spirit to the construction of a free product of vector spaces with speciﬁed state vectors from Subsection 1.1.2.

n ˚ o Construction 3.1.5. Let (Xι, Xι, pι) be B-B-bimodules with specified B-vector states. ι∈I n ˚ o For simplicity, let Eι denote EL(Xι). The free product of (Xι, Xι, pι) with amalgamation ι∈I over B is defined to be the B-B-bimodule with specified vector state (X , X˚, p) where X = B ⊕ X˚ and X˚ is the B-B-bimodule

˚ M M ˚ ˚ X = Xk1 ⊗B · · · ⊗B Xkn

n≥1 k16=k26=···6=kn

with the left and right actions of B on X˚ deﬁned by

b · (x1 ⊗ · · · ⊗ xn) = (Lbx1) ⊗ · · · ⊗ xn

(x1 ⊗ · · · ⊗ xn) · b = x1 ⊗ · · · ⊗ (Rbxn).

We use ∗ι∈I Xι to denote X . 74 For each ι ∈ I, we deﬁne the left representation λι : L`(Xι) → L(X ) as follows: let    M M ˚ ˚  Wι : X → Xι ⊗B B ⊕ Xk1 ⊗B · · · ⊗B Xkn    n≥1 k16=k26=···6=kn k16=k be the B-B-bimodule isomorphism deﬁned analogously to Wι from Subsection 1.1.2, and set

−1 λι(T ) = Wι (T ⊗ I)Wι.

Note that this is unambiguous precisely because T ∈ L`(Xι), so we have

(T ⊗ I)(x · b) ⊗ ξ = (T x) · b ⊗ ξ = (T ⊗ I)(x ⊗ b · ξ).

We can compute λι(T ) explicitly: for b ∈ B and T ∈ L`(Xι),

λι(T )(b) = Ek(T )b + (T − LEι(T ))b, while   Lp (T x )x2 ⊗ · · · ⊗ xn  k 1 ˚  if x1 ∈ Xι  + ([(1 − pk)T x1] ⊗ · · · ⊗ xn) λ (T )(x ⊗ · · · ⊗ x ) = . ι 1 n  LEk(T )x1 ⊗ · · · ⊗ xn  if x 6∈ X˚  1 ι  + [(T − LEk(T ))1B] ⊗ x1 ⊗ · · · ⊗ xn

Here as usual we interpret a tensor product of length zero as the vector 1B. Observe that λι is a homomorphism, λι(Lb) = Lb, and λι(L`(Xι)) ⊆ L`(X ).

Similarly, for each ι ∈ I, we deﬁne the map ρι : Lr(Xι) → L(X ) as follows: let    M M ˚ ˚  Uι : X → B ⊕ Xk1 ⊗B · · · ⊗B Xkn  ⊗B Xι   n≥1 k16=k26=···6=kn kn6=k be the B-B-bimodule isomorphism analogous to Uι in Subsection 1.1.2, and deﬁne

−1 ρι(T ) = Uι (I ⊗ T )Uι; again this is well-deﬁned precisely because T commutes with the left action of B. As before, we ﬁnd

ρι(T )(b) = bEι(T ) + (T − REι(T ))b, 75 and   x1 ⊗ · · · ⊗ Rp (T x )xn−1  k n ˚  if xn ∈ Xι  + (x1 ⊗ · · · ⊗ [(1 − pk)T xn]) ρ (T )(x ⊗ · · · ⊗ x ) = , ι 1 n  x1 ⊗ · · · ⊗ REι(T )xn  if x 6∈ X˚  n ι  + x1 ⊗ · · · ⊗ xn ⊗ [(T − REι(T ))1B] for all T ∈ L(Xι). Clearly ρι is a homomorphism, ρι(Rb) = Rb, and ρι(Lr(Xι)) ⊆ Lr(X ).

In addition, note that if T ∈ L`(Xι) then

EL(X )(λι(T )) = p(λι(T )1B) = p(Eι(T ) + [T − LEι(T )]1B) = Eι(T )

and similarly EL(X )(ρι(T )) = Eι(T ) if T ∈ Lr(Xι). Hence, the above shows that L(X ) contains each L(Xι) in a left-preserving, right-preserving, state-preserving way.

With computation, we see that λi(T ) and ρj(S) commute when T ∈ L`(Xi), S ∈ Lr(Xj), and i 6= j. Indeed, notice if b ∈ B then

λi(T )ρj(S)b = λi(T ) bEj(S) + (S − REj (S))b

= Ei(T )bEj(S) + (T − LEi(T ))bEj(S) + LEi(T )(S − REj (S))b + [(T − LEi(T ))1B] ⊗ [(S − REj (S))b] , whereas

ρj(S)λi(T )b = ρj(S) Ei(T )b + (T − LEi(T ))b

= Ei(T )bEj(S) + (S − REj (S))Ei(T )b + REj (S)(T − LEi(T ))b + [(T − LEi(T ))b] ⊗ [(S − REj (S))1B] .

Since T ∈ L`(Xi) and S ∈ Lr(Xj), one sees that

LEi(T )(S − REj (S))b = (S − REj (S))LEi(T )b = (S − REj (S))Ei(T )b,

REj (S)(T − LEi(T ))b = (T − LEi(T ))REj (S)b = (T − LEi(T ))bEj(S),

76 and

[(T − LEi(T ))b] ⊗ [(S − REj (S))1B] = [(T − LEi(T ))Rb1B] ⊗ [(S − REj (S))1B]

= [Rb(T − LEi(T ))1B] ⊗ [(S − REj (S))1B]

= [(T − LEi(T ))1B] ⊗ [Lb(S − REj (S))1B]

= [(T − LEi(T ))1B] ⊗ [(S − REj (S))Lb1B]

= [(T − LEi(T ))1B] ⊗ [(S − REj (S))b].

Thus λi(T )ρj(S)b = ρj(S)λi(T )b. Similar computations show λi(T ) and ρj(T ) commute on ˚ ˚ ˚ ˚ Xi, Xj, and Xi ⊗ Xj, and it is trivial to see that λi(T ) and ρj(T ) commute on all other components of X˚.

Note that λi(T ) and ρi(S) need not commute, though their commutator will be supported ˚ on B ⊕ Xi and there will be equal to the commutator [T,S].

3.1.2 Abstract structures for bi-free probability with amalgamation.

The purpose of this section is to develop an abstract notion of the pair (L(X ),EL(X )). Based on the previous section and Proposition 3.1.4, we make the following deﬁnition.

Deﬁnition 3.1.6. A B-B-non-commutative probability space is a triple (A,EA, ε) where A

op op is a unital algebra, ε : B ⊗ B → A is a unital homomorphism such that ε|B⊗1B and ε|1B⊗B are injective, and EA : A → B is a unital linear map such that

EA(ε(b1 ⊗ b2)T ) = b1EA(T )b2 for all b1, b2 ∈ B and T ∈ A, and

EA(T ε(b ⊗ 1B)) = EA(T ε(1B ⊗ b)) for all b ∈ B and T ∈ A.

In addition, we deﬁne the unital subalgebras A` and Ar of A by

A` := {T ∈ A | T ε(1B ⊗ b) = ε(1B ⊗ b)T for all b ∈ B}

Ar := {T ∈ A | T ε(b ⊗ 1B) = ε(b ⊗ 1B)T for all b ∈ B} . 77 We call A` and Ar the left and right algebras of A, respectively.

If (X , X˚, p) is a B-B-bimodule with a speciﬁed B-vector state, we see via Proposition

3.1.4 that (L(X ),EL(X ), ε) is a B-B-non-commutative probability space where EL(X ) is as

op in Deﬁnition 3.1.3 and ε : B ⊗ B → B is deﬁned by ε(b1 ⊗ b2) = Lb1 Rb2 . As such, in an arbitrary B-B-non-commutative probability space (A,EA, ε), we will often use Lb instead of

ε(b ⊗ 1) and Rb instead of ε(1 ⊗ b), in which case Lb ∈ A` and Rb ∈ Ar for all b ∈ B. For b ∈ B, we will call Lb a left B-operator and Rb a right B-operator.

It may appear that Deﬁnition 3.1.6 is incompatible with the notion of a B-probability space in free probability: that is, a pair (A, E) where A is a unital algebra containing B, and E : A → B is a linear map such that E(b1T b2) = b1E(T )b2 for all b1, b2 ∈ B and T ∈ A. However, A is a B-B-bimodule by left and right multiplication by B, and A can be made into a B-B-bimodule with speciﬁed B-vector state via p = E and X˚ = ker(E). Hence the above discussion implies L(A) is a B-B-non-commutative probability space with

EL(A)(T ) = E(T )

for all T ∈ L(A). In addition, we can view A as a unital subalgebra of both L`(A) and

Lr(A) by left and right multiplication on A respectively.

Viewing A ⊆ L`(A), it is clear we can recover the joint B-moments of elements of A from

EL(A). Indeed, for T ∈ A ⊆ L`(A) we have

EL(A)(Lb1 TLb2 ) = EL(A)(Lb1 TRb2 ) = EL(A)(Lb1 Rb2 T ) = b1EL(A)(T )b2, which is consistent with the deﬁning property of E. In particular, the same proof shows op (A`,E) is a B-non-commutative probability space and (Ar,E) is a B -non-commutative probability space.

One should note that Definition 3.1.6 differs slightly from [30, Definition 8.3]. How- ever, given Proposition 3.1.4 and the following result which demonstrates that a B-B-non- commutative probability space embeds into L(X ) for a B-B-bimodule with a specified B- vector state X , Definition 3.1.6 indeed specifies the correct abstract objects to study.

78 Theorem 3.1.7. Let (A,EA, ε) be a B-B-non-commutative probability space. Then there exists a B-B-bimodule with a speciﬁed B-vector state (X , X˚, p) and a unital homomorphism θ : A → L(X ) such that

θ(Lb1 Rb2 ) = Lb1 Rb2 , θ(A`) ⊆ L`(X ), θ(Ar) ⊆ Lr(X ), and EL(X )(θ(T )) = EA(T )

for all b1, b2 ∈ B and T ∈ A.

Proof. Consider the vector space X = B ⊕ Y, where

Y = ker(EA)/span {TLb − TRb | T ∈ A, b ∈ B} .

Note Y is a well-deﬁned quotient vector space since EA(TLb − TRb) = 0 by Deﬁnition 3.1.6. We will postpone describing the B-B-module structure on X until later in the proof.

Let q : ker(EA) → Y denote the canonical quotient map. Then, for T,A ∈ A with

EA(A) = 0 and b ∈ B, we deﬁne θ(T ) ∈ L(X ) by

θ(T )(b) = EA(TLb) ⊕ q(TLb − LEA(TLb))

and

θ(T )(q(A)) = EA(TA) ⊕ q(TA − LEA(TA)).

Note that span {TLb − TRb | T ∈ A, b ∈ B} is a left-ideal in A, so q(A) = 0 implies q(TA) = 0 and thus θ is well-deﬁned.

We wish to show that θ is a homomorphism; it is immediate that θ is linear. To see that θ is multiplicative, ﬁx T,S ∈ A. If b ∈ B, then

θ(T )(b) = EA(TLb) ⊕ q(TLb − LE(ATLb)).

Thus θ(S)(θ(T )(b)) = EA(SLE (TL )) ⊕ q SLE (TL ) − LE (SL ) A b A b A EA(TLb) + EA(S(TLb − LE (TL ))) ⊕ q S(TLb − LE (TL )) − LE (S(TL −L )) A b A b A b EA(TLb) = EA(STLb) ⊕ q STLb − LEA(STLb) = θ(ST )(b).

79 Similarly, if q(A) ∈ Y then

θ(T )(q(A)) = EA(TA) ⊕ q(TA − LEA(TA)).

Thus

θ(S)(θ(T )(q(A))) = EA SLE (TA) ⊕ q SLE (TA) − LE (SL ) A A A EA(TA)

+ EA(S(TA − LE (TA))) ⊕ q(S(TA − LE (TA)) − LE (S(TA−L ))) A A A EA(TA) = EA(STA) ⊕ q STA − LEA(STA) = θ(ST )(q(A)).

Hence θ is a homomorphism.

To make X a B-B-bimodule, we deﬁne

b · ξ = θ(Lb)(ξ) and ξ · b = θ(Rb)(ξ)

for all ξ ∈ X and b ∈ B; thus we automatically have θ(Lb1 Rb2 ) = Lb1 Rb2 .

To demonstrate that X is indeed a B-B-bimodule with a speciﬁed vector state, we must show that Y is invariant under this B-B-bimodule structure, and that the B-B-bimodule structure when restricted to B ⊆ X is the canonical one. If b, b0 ∈ B and q(A) ∈ Y, then

0 0 0 θ(L )(b ) = E (L L 0 ) ⊕ q(L L 0 − L ) = bb ⊕ q(L 0 − L 0 ) = bb ⊕ 0 b A b b b b EA(LbLb0 ) bb bb and

θ(Lb)(q(A)) = EA(LbA) ⊕ q(LbA − LEA(LbA))

= bEA(A) ⊕ q(LbA − LEA(LbA)) = 0 ⊕ q(LbA − LEA(LbA)).

Similarly,

0 0 0 θ(R )(b ) = E (R L 0 ) ⊕ q(R L 0 − L ) = b b ⊕ q(L 0 R − L 0 L ) = b b ⊕ 0 b A b b b b EA(RbLb0 ) b b b b

and

θ(Rb)(q(A)) = EA(RbA) ⊕ q(RbA − LE(RbA))

= EA(A)b ⊕ q(RbA − LEA(RbA)) = 0 ⊕ q(RbA − LEA(RbA)). 80 Thus X is a B-B-bimodule with a speciﬁed B-vector state.

Since θ is a homomorphism, it is clear that θ(A`) ⊆ L`(X ) and θ(Ar) ⊆ Lr(X ) due to the deﬁnition of the B-B-bimodule structure on X . Finally, if T ∈ A then

EL(X )(θ(T )) = p(θ(T )(1B ⊕ 0)) = p(EA(T ) ⊕ q(T − LEA(T ))) = EA(T ).

3.1.3 Bi-free families of pairs of B-faces.

With the notion of a B-B-non-commutative probability space from Definition 3.1.6, we are now able to define the main concept of this chapter, following [30, Definition 8.5].

Deﬁnition 3.1.8. Let (A,EA, ε) be a B-B-non-commutative probability space. A pair of B-faces of A is a pair (C,D) of unital subalgebras of A such that

op ε(B ⊗ 1B) ⊆ C ⊆ A` and ε(1B ⊗ B ) ⊆ D ⊆ Ar.

A family {(Cι,Dι)}ι∈I of pairs of B-faces of A is said to be bi-free with amalgamation over B (or simply bi-free over B) if there exist B-B-bimodules with speciﬁed B-vector states n ˚ o (Xι, Xι, pι) and unital homomorphisms lι : Cι → L`(Xι), rι : Dι → Lr(Xι) such that ι∈I the joint distribution of {(Cι,Dι)}ι∈I with respect to EA is equal to the joint distribution of the images {((λι ◦ lι)(Cι), (ρι ◦ rι)(Dι))}ι∈I inside L(∗ι∈I Xι) with respect to EL(∗ι∈I Xι).

It will be an immediate consequence of Theorem 3.5.4 that the selection of representations

in Deﬁnition 3.1.8 does not matter (see [30, Proposition 2.9]). Note that if {(Cι,Dι)}ι∈I is

bi-free over B, then {Cι}ι∈I is free with amalgamation over B (as is {Dι}ι∈I ) and Ci and Dj commute in distributions whenever i 6= j.

To conclude this section, we give the following example.

Example 3.1.9. Let (M1, τ1) and (M2, τ2) be II1 factors, and (N, τN ) a common von Neu-

mann sub-algebra. Then if M = M1 ∗N M2 is their amalgamated free product as von Neumann algebras, L2(M) has the structure of an N-N-bimodule via left and right multiplication. If we take p to be the orthogonal projection of L2(M) onto L2(N), this makes

2 2 ⊥ (L (M),L (N) , p) into a an N-N-bimodule with speciﬁed B-vector state. Then taking λi 81 and ρi to be the left and right representations of Mi on M, we ﬁnd that (λ1(M1), ρ1(M1))

2 and (λ2(M2), ρ2(M2)) are bi-free with amalgamation over N in L (L (M)).

3.2 Operator-valued bi-multiplicative functions.

In this section, we will develop a notion of B-valued bi-multiplicative functions in order to study B-B-non-commutative probability spaces (compare [16, Section 2] or [22, Section 2]). Our goal is once again to use this theory to understand operator-valued bi-free cumulants.

3.2.1 Deﬁnition of bi-multiplicative functions.

We begin by examining the operator-valued generalization of the multiplicative functions used in Chapter 2.

Deﬁnition 3.2.1. Let (A, E, ε) be a B-B-non-commutative probability space and let

[ [ Φ: BNC(χ) × Aχ(1) × · · · × Aχ(n) → B n≥1 χ:[n]→{`,r}

be a function that is linear in each Aχ(k). We say that Φ is bi-multiplicative if for every

χ :[n] → {`, r}, Tk ∈ Aχ(k), b ∈ B, and π ∈ BN C(χ), the following four conditions hold:

(i) Let q = max {k ∈ [n] | χ(k) 6= χ(n)} .

If χ(n) = ` then   Φ1χ (T1,...,Tq−1,TqRb,Tq+1,...,Tn) if q 6= −∞ Φ1χ (T1,...,Tn−1,TnLb) = .

 Φ1χ (T1,...,Tn−1,Tn)b if q = −∞

If χ(n) = r then   Φ1χ (T1,...,Tq−1,TqLb,Tq+1,...,Tn) if q 6= −∞ Φ1χ (T1,...,Tn−1,TnRb) = .

 bΦ1χ (T1,...,Tn−1,Tn) if q = −∞

82 (ii) Let p ∈ [n], and let

r = max {k ∈ [n] | χ(k) = χ(p), k < p} .

If χ(p) = ` then

Φ1χ (T1,...,Tp−1,LbTp,Tp+1,...,Tn)   Φ1 (T1,...,Tq−1,TqLb,Tq+1,...,Tn) if q 6= −∞ = χ .

 bΦ1χ (T1,T2,...,Tn) if q = −∞

If χ(p) = r then

Φ1χ (T1,...,Tp−1,RbTp,Tp+1,...,Tn)   Φ1 (T1,...,Tq−1,TqRb,Tq+1,...,Tn) if q 6= −∞ = χ .

 Φ1χ (T1,T2,...,Tn)b if q = −∞

(iii) Suppose that V1,...,Vm are χ-intervals which partition [n] so that π ≤ {V1,...,Vm}.

Further, suppose V1 ≺χ ... ≺χ Vm (i.e., the relation ≺χ holds for every choice of elements from these sets). Then

Φ (T ,...,T ) = Φ ((T ,...,T )| ) ··· Φ ((T ,...,T )| ) . π 1 n π|V1 1 n V1 π|Vm 1 n Vm

(iv) Suppose that V and W partition [n], π ≤ {V,W }, and V is a χ-interval which is inner in {V,W } in the sense of Subsection 2.5.2. Let θ := max k ∈ W | k ≺χ min(V ) , γ := min k ∈ W | max(V ) ≺χ k . ≺χ ≺χ ≺χ ≺χ

Then

Φπ(T1,...,Tn)  Φπ| T1,...,Tθ−1,TθLΦ ((T ,...,T )| ),Tθ+1,...,Tn |W if χ(θ) = `  W π|V 1 n V =  Φπ| T1,...,Tθ−1,RΦ ((T ,...,T )| )Tθ,Tθ+1,...,Tn |W if χ(θ) = r W π|V 1 n V  Φπ| T1,...,Tγ−1,LΦ ((T ,...,T )| )Tγ,Tγ+1,...,Tn |W if χ(γ) = `  W π|V 1 n V = .  Φπ| T1,...,Tγ−1,TγRΦ ((T ,...,T )| ),Tγ+1,...,Tn |W if χ(γ) = r W π|V 1 n V 83 Example 3.2.2. Suppose that Φ is a bi-multiplicative function, and that χ : [5] → {`, r} corresponds to the sequence (`, `, r, `, r). Using Properties (i) and (ii), we obtain that

Φπ(T1,Lb1 T2,Rb2 T3,T4,T5Rb3 ) = Φπ(T1Lb1 ,T2,T3,T4Lb3 ,T5)b2.

This can be thought of as allowing us to move elements of B between nodes on the diagram of the corresponding bi-non-crossing partition:

Rb2 T1 T1 Lb1 Lb1 T2 T2 Rb2 T3 = T3 T4 T4 Lb3 T5 T5 Rb3

Example 3.2.3. Again, take Φ to be a bi-multiplicative function. Suppose χ : [8] → {`, r} corresponds to the sequence (`, r, `, `, `, r, r, `). Let π = {{1, 3, 4} , {5, 7, 8} , {2, 6}} and σ = {{1, 2, 6} , {3, 7} , {4, 5, 8}}. Then

Φπ(T1,...,T8) = Φπ1 (T1,T3,T4)Φπ2 (T5,T7,T8)Φπ3 (T2,T6),

and Φ (T ,...,T ) = Φ T L ,T ,T . σ 1 8 σ1 1 Φσ T3,T7R 2 6 2 ( Φσ3 (T4,T5,T6))

T1 T1 T2 T2 T3 T3 T4 T4 T5 T5 T6 T6 T7 T7 T8 T8 π σ

The underlying idea is this: any interval of blocks in π may be evaluated with Φ and replaced by the corresponding element of B at the location in the diagram where the block was removed. Only blocks which do not bound other blocks may be reduced in this manner; inner blocks must be reduced ﬁrst.

84 Although Deﬁnition 3.2.1 is cumbersome (due to the necessity of specifying cases based on whether certain terms are left or right operators), its properties can be viewed as direct analogues of those of a multiplicative map as described in [16, Section 2.2]. Indeed,

for π ∈ BN C(χ) and a bi-multiplicative map Φ, each expression of Φπ(T1,...,Tn) in Def-

−1 inition 3.2.1 comes from viewing sχ ◦ π ∈ NC(n), rearranging the n-tuple (T1,...,Tn) to

(Tsχ(1),...,Tsχ(n)), replacing any occurrences of LbTj, TjLb, RbTj, and TjRb with bTj, Tjb,

Tjb, and bTj respectively, applying one of the properties of a multiplicative map from [16, Sec- tion 2.2], and reversing the above identiﬁcations. In particular, these properties reduce to those of a multiplicative map when χ−1({`}) = [n]. We use the more complex Deﬁnition 3.2.1 as it will be easier to verify for functions later on.

Since a bi-multiplicative function satisﬁes all of these properties, it is easy to see that if

Φ is bi-multiplicative, then Φπ(T1,...,Tn) is determined by the values

n 0 o Φ1χ0 (S1,...,Sm) | m ∈ N, χ : {1, . . . , m} → {`, r} ,Sk ∈ Aχ(k) .

There may be multiple ways to reduce Φ to an expression involving elements from the above set, but Deﬁnition 3.2.1 implies that all such reductions are equal.

Note that Deﬁnition 3.2.1 automatically implies additional properties for bi-multiplicative functions. Indeed one can either verify the following proposition via Deﬁnition 3.2.1 and casework, or can appeal to the fact that the properties of bi-multiplicative functions can be described via the properties of multiplicative functions as above, and use the fact that multiplicative functions have additional properties (see, e.g., [22, Remark 2.1.3]).

Proposition 3.2.4. Let (A, E, ε) be a B-B-non-commutative probability space and let

[ [ Φ: BNC(χ) × Aχ(1) × · · · × Aχ(n) → B n≥1 χ:[n]→{`,r} be a bi-multiplicative function. Given any χ :[n] → {`, r}, π ∈ BN C(χ), and Tk ∈ Aχ(k)

Properties (i) and (ii) of Deﬁnition 3.2.1 hold when 1χ is replaced with π.

85 3.3 Bi-free operator-valued moment function is bi-multiplicative.

In this section, we will deﬁne the bi-free operator-valued moment function based on re-

cursively deﬁned functions Eπ(T1,...,Tn) that appear via actions on free product spaces. However, it is not immediate that it is bi-multiplicative. The proof of this result requires substantial case work, to which this section is dedicated.

3.3.1 Deﬁnition of the bi-free operator-valued moment function.

We will begin with the recursive deﬁnition of expressions that appear in the operator-valued moment polynomials. These will arise in the proof of Theorem 3.5.4, where we will give a characterisation of bi-freeness with amalgamation over B akin to that of bi-freeness in Corollary 2.4.6.

Deﬁnition 3.3.1. Let (A, E, ε) be a B-B-non-commutative probability space. For χ :[n] →

{`, r}, π ∈ BN C(χ), and T1,...,Tn ∈ A, we deﬁne Eπ(T1,...,Tn) ∈ B via the following recursive process. Let V be the block of π that terminates closest to the bottom, so min(V ) is largest among all blocks of π. Then:

• If π contains exactly one block (that is, π = 1χ), we set E1χ (T1,...,Tn) = E(T1 ··· Tn).

• If V = {k + 1, . . . , n} for some k < n, then min(V ) is not adjacent to any spines of π and we define   Eπ| c (T1,...,TkLE (T ,...,T )) if χ(min(V )) = ` V π|V k+1 n Eπ(T1,...,Tn) := .  Eπ| c (T1,...,TkRE (T ,...,T )) if χ(min(V )) = r V π|V k+1 n In the long run, it will not matter if we choose L or R by the first part of this recursive definition and Definition 3.1.6.

• Otherwise, min(V ) is adjacent to a spine. Let W denote the block of π corresponding to the spine adjacent to min(V ), and let k be the ﬁrst element of W below where V terminates – that is, k is the smallest element of W that is larger than min(V ). We

Notice that if B = C and E = ϕ is a state, then Eπ in the above sense is precisely ϕπ in the notation from Chapter 2.

Example 3.3.2. Let π be the following bi-non-crossing partition. 1 2 3 4 5 6 7 8 9

Then Eπ(T1,...,T9) = E T1T2LE T L T R T ( 3 E(T4T7) 5 E(T6T8) 9) via the following sequence of diagrams (where X = LE T L T R T ): ( 3 E(T4T7) 5 E(T6T8) 9)

T1 T1 T1 T1 T2 T2 T2 T2 X T3 T3 T3 T4 T4 LE(T4T7) T5 T5 T5 T6 T7 T7 T8 RE(T6T8) RE(T6T8) T9 T9 T9

Note that the definition of Eπ(T1,...,Tn) is invariant under B-B-non-commutative probability space embeddings, such as those listed in Theorem 3.1.7. Observe that in the context of Definition 3.3.1, we ignore the notions of left and right operators. However, we are ulti- mately interested in the following. 87 Definition 3.3.3. Let (A, E, ε) be a B-B-non-commutative probability space. The bi-free operator-valued moment function [ [ E : BNC(χ) × Aχ(1) × · · · × Aχ(n) → B n≥1 χ:[n]→{`,r} is defined by

Eπ(T1,...,Tn) = Eπ(T1,...,Tn) for each χ :[n] → {`, r}, π ∈ BN C(χ), and Tk ∈ Aχ(k).

Our next goal is the prove the following which is not apparent from Deﬁnition 3.3.1.

Theorem 3.3.4. The operator-valued bi-free moment function E on A is bi-multiplicative.

We divide the proof of the above theorem into several lemmata, verifying various of properties from Deﬁnition 3.2.1. Properties (i) and (ii) are immediate but, unfortunately, the remaining properties are not as easily veriﬁed.

Lemma 3.3.5. The operator-valued bi-free moment function E satisﬁes Properties (i) and (ii) of Deﬁnition 3.2.1.

Proof. This follows from the facts that E1χ (T1,...,Tn) = E(T1 ··· Tn), that left (resp. right) variables commute with Rb (resp. Lb) for b ∈ B, and that E(T1 ··· TnLb) = E(T1 ··· TnRb).

3.3.2 Veriﬁcation of Property (iii) from Deﬁnition 3.2.1 for E.

Lemma 3.3.6. The operator-valued bi-free moment function E satisﬁes Property (iii) of Deﬁnition 3.2.1.

Proof. We claim it suﬃces to consider the case when σ = {V1,...,Vm} is the ﬁnest partition such that each Vj is a χ-interval and π ≤ σ; indeed, given any other ρ with π ≤ ρ having

χ-intervals W1 ≺χ · · · ≺χ Wm0 as blocks, applying the argument for this restricted case to the blocks of ρ yields

Eπ| ((Ti)|W1 ) ···Eπ| ((Ti)|W 0 ) = Eπ| ((Ti)|V1 ) ···Eπ| ((Ti)|Vm ) = Eπ(T1,...,Tn). W1 Wm0 m V1 Vm 88 Note that for σ above, it must be the case that min≺χ (Vi) ∼π max≺χ (Vi), as otherwise a ﬁner partition is possible.

We now proceed by induction on m, the number of blocks in σ, with the case m = 1 being immediate. Assume Property (iii) of Deﬁnition 3.2.1 is satisﬁed for E for all smaller values

of m. Fix V1,...,Vm and note that either 1 ∈ V1 (i.e. χ(1) = `) or 1 ∈ Vm (i.e. χ(1) = r).

0 We will treat the case when 1 ∈ V1; for the other case, consult a mirror. Let V1 ⊆ V1 be the block of π containing 1 and max≺χ (V1). The proof is divided into three cases.

Case 1: min(Vk) > max(V1) for all k 6= 1. As an example of this case, consider the following diagram where V1 = {1, 2, 3}, V2 = {4, 6}, and V3 = {5, 7, 8, 9}.

1 2 3 4 5 6 7 8 9

In this case, drawing a horizontal line directly beneath max(V1) will hit no spines in π and −1 0 Sm V1 ⊆ χ ({`}). Let V1 = {1 = q1 < q2 < ··· < qp} and V0 = k=2 Vk. Repeatedly applying

the deﬁnition of E, we may ﬁnd b1, . . . , bp−1 ∈ B depending only on (T1,...,Tn)|V1 and π|V1 , so that writing T 0 = T L we have qk qk bk

0 0 0 Eπ(T1,...,Tn) = E Tq1 Tq2 ··· Tqp−1 Tqp LEπ| ((T1,...,Tn)|V ) V0 0 0 0 0 = E Tq1 Tq2 ··· Tqp−1 Tqp REπ| ((T1,...,Tn)|V ) . V0 0

0 By the assumptions in this case, each Tk ∈ A` for all k ∈ V1 and since right B-operators

89 commute with elements of A`, we obtain

0 0 0 Eπ(T1,...,Tn) = E Tq1 Tq2 ··· Tqp REπ| ((T1,...,Tn)|V ) V0 0 0 0 0 = E REπ| ((T1,...,Tn)|V )Tq1 Tq2 ··· Tqp V0 0 = E(T 0 T 0 ··· T 0 )E ((T ,...,T )| ) q1 q2 qp π|V0 1 n V0 = E ((T ,...,T )| )E ((T ,...,T )| ) π|V1 1 n V1 π|V0 1 n V0 = E ((T ,...,T )| ) E ((T ,...,T )| ) ···E ((T ,...,T )| ) π|V1 1 n V1 π|V2 1 n V1 π|Vm 1 n Vm with the last step following by the inductive hypothesis.

If we are not in Case 1, then there exists a k 6= 1 such that min(Vk) < max(V1). In particular, Vm must terminate on the right above max(V1), so min(Vm) < max(V1) and

χ(min(Vm)) = r. We thus ﬁnd that there are two further cases.

Case 2: max(V1) < max(Vm). As an example of this case, consider the following diagram where V1 = {1, 3}, V2 = {4, 6, 7, 8, 9}, and V3 = {2, 5}.

1 2 3 4 5 6 7 8 9

−1 Again V1 ⊆ χ ({`}), as its lowest element is higher than the lowest element of Vm. With the same conventions as above, by repeated application of the rules deﬁning E, we obtain

0 0 0 0 0 Eπ(T1,...,Tn) = E Tq1 Tq2 ··· Tqp REπ| ((T1,...,Tn)|V )Tqp +1 ··· Tqp , 1 V0 0 1 2

0 where p1 is the smallest element of V1 greater than min(Vm). Note that all of the other blocks of π are consumed into the block Vm: blocks are only consumed into either the lowest block or a block they are tangled with, and not block except Vm may be tangled with V1.

90 0 By the assumptions in this case, each Tk ∈ A` for all k ∈ V1 , and since right B-operators commute with elements of A`, one obtains

0 0 0 0 0 Eπ(T1,...,Tn) = E Tq1 Tq2 ··· Tqp REπ| ((T1,...,Tn)|V )Tqp +1 ··· Tqp 1 V0 0 1 2 0 0 0 = E REπ| ((T1,...,Tn)|V )Tq1 Tq2 ··· Tqp V0 0 2 0 0 0 = E(T T ··· T )Eπ| ((T1,...,Tn)|V ) q1 q2 qp2 V0 0 = E ((T ,...,T )| )E ((T ,...,T )| ) π|V1 1 n V1 π|V0 1 n V0 = E ((T ,...,T )| ) E ((T ,...,T )| ) ···E ((T ,...,T )| ) , π|V1 1 n V1 π|V2 1 n V1 π|Vm 1 n Vm with the last step following by the inductive hypothesis.

Case 3: max(V1) > max(Vm). As an example of this case, consider the following diagram where V1 = {1, 5}, V2 = {6, 8, 9}, V3 = {4, 7}, and V4 = {2, 3}.

1 2 3 4 5 6 7 8 9

Sm−1 Let V0 = k=1 Vk. Once again appealing to the properties of E given in Definition 3.3.1 0 0 0 we may find T ,...,T and S ∈ A where T differs from Tk by a left multiplication operator, q1 qp1 k so that

0 0 0 Eπ(T1,...,Tn) = E Tq Tq ··· Tq REπ| ((T1,...,Tn)|V )S . 1 2 p1 Vm m

91 Since right B-operators commute with elements of A`, one obtains

0 0 0 Eπ(T1,...,Tn) = E Tq Tq ··· Tq REπ| ((T1,...,Tn)|V )S 1 2 p1 Vm m

0 0 0 = E REπ| ((T1,...,Tn)|V )Tq Tq ··· Tq S Vm m 1 2 p1

0 0 0 = E(T T ··· T S)Eπ| ((T1,...,Tn)|V ) q1 q2 qp1 Vm m = E ((T ,...,T )| )E ((T ,...,T )| ) π|V0 1 n V0 π|Vm 1 n Vm = E ((T ,...,T )| ) E ((T ,...,T )| ) ···E ((T ,...,T )| ) π|V1 1 n V1 π|V2 1 n V1 π|Vm 1 n Vm with the last step following by the inductive hypothesis.

3.3.3 Veriﬁcation of Property (iv) from Deﬁnition 3.2.1 for E.

We begin with the following intermediate step on the way to verifying that E satisﬁes Prop- erty (iv). Recall that in the context of Deﬁnition 3.2.1 Property (iv), we have an inner χ-interval V , W := [n] \ V , and we have labelled the nodes which are immediately before and after V in the ≺χ-order as θ and γ, respectively.

Lemma 3.3.7. The operator-valued bi-free moment function E satisﬁes Property (iv) of

Deﬁnition 3.2.1 with the additional assumption that there exists a block W0 ⊆ W of π such that

θ, γ, min([n]), max([n]) ∈ W0. ≺χ ≺χ

Proof. We will present only the proof of the case χ(θ) = ` as the other case is similar.

Let {V1 ≺χ ... ≺χ Vm} be the ﬁnest partition of V consisting of χ-intervals which has

π|V as a reﬁnement. Note that θ immediately precedes min≺χ (V1) and γ immediately follows max≺χ (Vm).

The proof is now divided into three cases. In the case χ(n) = `, we have χ ≡ ` since q = −∞.

92 Case 1: χ(γ) = `. As an example of this case, consider the following diagram where

W = W0 = {1, 5, 9}, V1 = {2, 3}, V2 = {4, 6, 7, 8}, θ = 1, and γ = 9.

1 2 3 4 5 6 7 8 9

−1 In this case V ⊆ χ ({`}). Write Xk = LEπ| ((T1,...,Tn)|V ) and denote the elements of W0 Vk k by q1 < q2 < ··· < qkm+1 . Then

0 0 0 0 0 0 0 Eπ(T1,...,Tn) = E T T ··· T X1T ··· T XmT ··· T , q1 q2 qk1 qk1+1 qkm qkm+1 qkm+1

0 where Tk is Tk, potentially multiplied on the left and/or right by appropriate Lb and Rb.

Here Tθ appears left of X1, γ = qkm+1 , and every operator between Tθ and Tγ is either some

Xk or a right operator. Hence, by the commutation of left B-operators with elements of Ar, we obtain

0 0 0 0 Eπ(T1,...,Tn) = E T ··· T RbLb0 (TθX1X2 ··· Xm) Rb00 T ··· T q1 qj−1 qj+1 qkm+1 for some b, b0, b00 ∈ B. Since

E ((T ,...,T )| ) ···E ((T ,...,T )| ) = E ((T ,...,T )| ), π|V1 1 n V1 π|Vm 1 n Vm π|V 1 n V by Lemma 3.3.6, we have

0 0 0 0 0 00 Eπ(T1,...,Tn) = E T ··· T RbLb TθLE ((T1,...,Tn)| ) Rb T ··· T q1 qj−1 π|V V qj+1 qkm+1

= Eπ|W T1,...,Tθ−1,TθLEπ| ((T1,...,Tn)|V ),Tθ+1,...,Tn V W

where the last step follows as Eπ|W ignores arguments corresponding to V .

93 Case 2: χ(γ) = r and θ < γ. As an example of this case, consider the following diagram where W = W0 = {1, 3, 6}, V1 = {2, 4}, V2 = {5, 8}, V3 = {7, 9}, θ = 1, and γ = 6.

1 2 3 4 5 6 7 8 9

Let p be the index of the last sub-interval of V to begin above γ, if such exists, and 0 otherwise. That is, let p be such that min(Vk) > γ if and only if k > p. Note that if k < p

−1 we have Vk ⊂ χ (`), and if p > 0, χ(min(Vp)) = `.

Considering the process which reductions are performed in evaluating E, we ﬁnd that every block of π contained in Vk with k > p will wind up either attached to a node in Vp or multiplied on the right of Tγ, depending on whether Vp terminates above or below γ. Letting

0 W0 = {q1 < ··· < qt}, we ﬁnd that there are Tk which diﬀer from Tk by multiplication by Lb and/or Rb (these corresponding to the reduction of other blocks in W ) so that Eπ(T1,...,Tn) is of the form 0 0 0 E Tq1 ··· TθZTγLY , where: Z is a product of T 0 having only right pieces (with θ < q < q < γ), L ((T )| qj j j+1 Eπ| i Vk Vk with k ≤ p, and possibly one term of the form L ; and Y is either L Eπ| ((Ti)|V ) Eπ| ((Ti)|V>p ) V≥p ≥p V>p or 1. The point, though, is that we can commute all the terms arising from V next to each other next to Tθ, and use Lemma 3.3.6 to replace their product by LE ((T )| ): π|V i V

0 0 0 0 0 Eπ(T1,...,Tn) = E(T ··· T T LE ((T )| )T ··· T ) q1 qj−1 θ π|V i V qj+1 γ = Eπ| T1,...,Tθ−1,TθLE ((T )| ),Tθ+1,...,Tn |W , W π|V i V where the second equality follows by reversing the sequence of reductions which compressed the blocks of W and created the T 0 , as these reductions could not be inﬂuenced by the qi presence or absences of V , being separated from it by W0. 94 Case 3: χ(γ) = r and θ > γ. The argument in this case is essentially the same as the above, except one ﬁnds that terms from V are collected as right multiplication operators

rather than as left ones, and always occurring after Tγ in the expansion of the product. All

things arising from W after Tγ must be left operators, though, and so they commute with the terms coming from the reduction of V which can the be collected as a right multiplication operator after Tθ. Since Tθ must be the last operator in W , we are able to replace this right multiplication coming from V with a left one, as required by Deﬁnition 3.2.1:

0 0 0 0 Eπ(T1,...,Tn) = E(T ··· T RE ((T )| )) = E(T ··· T LE ((T )| )). q1 θ π|V i V q1 θ π|V i V

In order to establish that Property (iv) holds without our special assumption above, it will be useful to prove the following stronger versions of Properties (i) and (ii).

Lemma 3.3.8. The operator-valued bi-free moment function E satisﬁes Property (i) of Def-

inition 3.2.1 when χ is constant and 1χ is replaced by an arbitrary π ∈ BN C(χ).

Proof. We will demonstrate the case χ ≡ `, as the other case follows mutatis mutandis. Notice that χ being constant means that q := max {k ∈ [n]|χ(k) 6= χ(n)} = −∞. Let

{V1 ≺χ ... ≺χ Vm} be the ﬁnest partition of V consisting of χ-intervals which has π|V as a

0 reﬁnement, and let Vi be the outer block of π contained in Vi. By Lemma 3.3.6, we may assume m = 1.

0 Writing V1 = {1 = q1 < q2 < ··· < qp+1 = n}, for some bj ∈ B depending only on π and

on (T ,...,T )| 0 c , 1 n (V1 ) Eπ(T1,...,Tn) = E Tq1 Lb1 Tq2 Lb2 ··· Tqp Lbp Tqp+1 .

Hence, by the commutation of right B-operators with elements of A`, we obtain

Eπ(T1,...,Tn)b = E Tq1 Lb1 Tq2 Lb2 ··· Tqp Lbp Tqp+1 b = E RbTq1 Lb1 Tq2 Lb2 ··· Tqp Lbp Tqp+1 = E Tq1 Lb1 Tq2 Lb2 ··· Tqp Lbp Tqp+1 Rb = E Tq1 Lb1 Tq2 Lb2 ··· Tqp Lbp Tqp+1 Lb

= Eπ(T1,...,TnLb). 95 Lemma 3.3.9. The operator-valued bi-free moment function E satisﬁes Property (ii) of

Deﬁnition 3.2.1 when q in the context of that deﬁnition is −∞, and 1χ is replaced by an arbitrary π ∈ BN C(χ).

Proof. We will assume χ(p) = ` as the case where χ(p) = r is once again similar. Let

{V1 ≺χ ... ≺χ Vm} be the ﬁnest partition of V consisting of χ-intervals which has π|V as a

0 reﬁnement, and let Vi be the outer block of π contained in Vi. Notice that since p is the ﬁrst 0 node on the left side of the partition, we necessarily have p ∈ V1 . Thus Lemma 3.3.6 implies we may reduce to the case where m = 1.

Notice that any block which will be contracted on the left in the evaluation of Eπ must

0 be below p; then writing V1 = {q1 < q2 < ··· < qk}, for some bj ∈ B depending only on

(T ,...,T )| 0 c and on π, for some S ∈ A, and for some z < k, 1 n (V1 )

Eπ(T1,...,Tn) = E (Tq1 Rb1 ··· Tqz Rbz TpS) .

Hence, by the commutation of left B-operators with elements of Ar, we obtain

0 bEπ(T1,...,Tn) = bE Tq1 Rb1 ··· Tqz Rbz TpS

= E (LbTq1 Rb1 ··· Tqz Rbz TpS)

= E (Tq1 Rb1 ··· Tqz Rbz LbTpS)

= Eπ(T1,...,Tp−1,LbTp,Tp+1,...,Tn).

Lemma 3.3.10. The operator-valued bi-free moment function E satisﬁes Property (iv) of Deﬁnition 3.2.1.

Proof. Again, only the proof of the ﬁrst case where χ(θ) = ` will be presented. We proceed by induction on the number of blocks U ∈ π with

U ⊆ W, min(U) ≺χ min(V ), and max(V ) ≺χ max(U), ≺χ ≺χ ≺χ ≺χ which we will denote by m. Such blocks are the ones which enclose V .

We will ﬁrst treat the case m = 0. Let

W1 = {k ∈ [n] | k χ θ} and W2 = {k ∈ [n] | γ χ k} . 96 Now both W1 and W2 are χ-intervals that are unions of blocks of π such that W = W1 t W2,

−1 and W1 ⊆ χ (`). Therefore by Lemmata 3.3.6 and 3.3.8,

E (T ,...,T ) = E ((T ,...,T )| )E ((T ,...,T )| )E ((T ,...,T )| ) π 1 n π|W1 1 n W1 π|V 1 n V π|W2 1 n W2

= Eπ| ((T1,...,Tθ−1,TθLE ((T ,...,T )| ))|W )Eπ| ((T1,...,Tn)|W ) W1 π|V 1 n V 1 W2 2

= Eπ| ((T1,...,Tθ−1,TθLE ((T ,...,T )| ),Tθ+1,...,Tn)|W ). W π|V 1 n V Note that we would either invoke Lemma 3.3.9 instead of 3.3.8 in the case χ(θ) = r, or else

bundle Eπ|V ((Ti)|V ) into the expectation corresponding to W2.

We must also establish the case m = 1 before we can begin the inductive step. Let W0 be the corresponding block counted by m, and let

α1 = min(W0), α2 = max({k ∈ W0 | k χ θ}), ≺χ ≺χ

β1 = max(W0), and β2 = min({k ∈ W0 | γ χ k}). ≺χ ≺χ Furthermore, let

0 0 W1 = {k ∈ [n] | k ≺χ α1} ,W2 = {k ∈ [n] | β1 ≺χ k} ,

00 00 W1 = {k ∈ [n] | α2 ≺ k χ θ} , and W2 = {k ∈ [n] | γ χ k ≺χ β2} .

Representing things graphically,

0 W1 0 W2 α1 β1 . . . . β2 α2 W 00 W 00 1 θ 2 γ

Therefore, if

0 0 0 0 X1 = Eπ| 0 ((T1,...,Tn)|W ),X2 = Eπ| 0 ((T1,...,Tn)|W ), W1 1 W2 2 00 00 00 00 X1 = Eπ| 00 ((T1,...,Tn)|W ), and X2 = Eπ| 00 ((T1,...,Tn)|W ), W1 1 W2 2 97 then by Lemmata 3.3.6 and 3.3.7,

Eπ(T1,...,Tn)

0 0 00 00 = X1Eπ| 00 00 ((T1,...,Tn)|W0∪W ∪W ∪V )X2 W0∪W1 ∪W2 ∪V 1 2 = X0 E T ,...,T ,T L ,T ,...,T | X0 1 π|W0 1 α2−1 α2 Eπ| 00 00 ((T1,...,Tn)|W 00∪W 00∪V ) α2+1 n W0 2 W1 ∪W2 ∪V 1 2

0 0 = X Eπ| T1,...,Tα −1,Tα LX00E ((T ,...,T )| )X00 ,Tα +1,...,Tn |W X . 1 W0 2 2 1 π|V 1 n V 2 2 0 2

00 If W1 is empty, then α2 = θ and

Tα LX00E ((T ,...,T )| )X00 = TθLE ((T ,...,T )| )LX00 . 2 1 π|V 1 n V 2 π|V 1 n V 2

00 On the other hand, if W1 is non-empty, then Lemma 3.3.8 implies that

00 00 X1 Eπ|V ((T1,...,Tn)|V ) = Eπ| 00 ((T1,...,Tθ−1,TθLEπ| ((T1,...,Tn)|V ),Tθ+1,...,Tn)|W ). W1 V 1

The result follows now from Lemmata 3.3.6 and 3.3.7 in the direction opposite the above.

Inductively, suppose that the result holds when the number of enclosing blocks is at

most m. Suppose W contains blocks W0,...,Wm of π which satisfy the above enclosing inequalities, ordered so that

min(W0) ≺χ · · · ≺χ min(Wm). ≺χ ≺χ

Note that as π is bi-non-crossing, this implies

max(Wm) ≺χ · · · ≺χ max(W0). ≺χ ≺χ

0 0 0 0 Let α1, α2, β1, β2, W1,W2,X1, and X2 be as above. Hence applying Lemmata 3.3.6 and 3.3.7 once again gives

Eπ(T1,...,Tn)

0 0 0 0 c = X1Eπ| 0 0 c ((T1,...,Tn)|(W ∪W ) )X2 (W1∪W2) 1 2 = X0 E ((T ,...,T ,T L ,T ,...,T ) | ) X0 , 1 π|W0 1 α2−1 α2 Z α2+1 n W0 2

98 0 0 c where Z := Eπ| 0 0 c ((T1,...,Tn)|(W0∪W ∪W ) ). Now, by the inductive hypothesis, we (W0∪W1∪W2) 1 2 see that

0 0 c Eπ| 0 0 c ((T1,...,Tn)|(W0∪W ∪W ) ) (W0∪W1∪W2) 1 2

0 0 c = Eπ| 0 0 c ((T1,...,Tθ−1,TθLEπ| ((T1,...,Tn)|V ),Tθ+1,...,Tn)|(W0∪W ∪W ) \V ). (W0∪W1∪W2) \V V 1 2

Hence, by substituting this expression into the above computation and applying Lemmata 3.3.6 and 3.3.7 in the opposite order, the inductive step is complete.

With this result, the proof of Theorem 3.3.4 is complete.

3.4 Operator-valued bi-free cumulants.

3.4.1 Cumulants via convolution.

Following [22, Deﬁnition 2.1.6], we begin with a deﬁnition of operator-valued convolution.

Deﬁnition 3.4.1. Let (A, E, ε) be a B-B-non-commutative probability space, let

[ [ Φ: BNC(χ) × Aχ(1) × · · · × Aχ(n) → B, n≥1 χ:[n]→{`,r} and let f ∈ IA(BNC). We deﬁne the convolution of Φ and f, denoted Φ ∗ f, by

X (Φ ∗ f)π(T1,...,Tn) := Φσ(T1,...,Tn)f(σ, π) σ∈BN C(χ) σ≤π for all χ :[n] → {`, r}, π ∈ BN C(χ), and Tk ∈ Aχ(k).

One can check that if Φ is as above and f, g ∈ IA(BNC) then (Φ ∗ f) ∗ g = Φ ∗(f ∗ g); this comes down to swapping the order of two ﬁnite sums.

Deﬁnition 3.4.2. Let (A, E, ε) be a B-B-non-commutative probability space and let E be the operator-valued bi-free moment function on A. The operator-valued bi-free cumulant function is the function

[ [ κ : BNC(χ) × Aχ(1) × · · · × Aχ(n) → B n≥1 χ:[n]→{`,r} 99 deﬁned by

κ := E ∗ µBNC.

Note for χ :[n] → {`, r}, π ∈ BN C(χ), and Tk ∈ Aχ(k) that

X X κπ(T1,...,Tn) = Eσ(T1,...,Tn)µBNC(σ, π) and Eπ(T1,...,Tn) = κσ(T1,...,Tn). σ≤π σ≤π

Also observe that if B = C is the complex numbers, the operator-valued bi-free cumulant function κ is precisely the usual bi-free cumulant functional.

3.4.2 Convolution preserves bi-multiplicativity.

It is now straightforward to demonstrate the operator-valued bi-free cumulant function is bi-multiplicative, as a corollary to the following theorem:

Theorem 3.4.3. Let (A, E, ε) be a B-B-non-commutative probability space, let

[ [ Φ: BNC(χ) × Aχ(1) × · · · × Aχ(n) → B, n≥1 χ:[n]→{`,r} and let f ∈ IA(BNC). If Φ is bi-multiplicative and f is multiplicative, then Φ ∗ f is bi- multiplicative.

Proof. Clearly (Φ ∗ f)π is linear in each entry. Furthermore, Proposition 3.2.4 establishes that Φ ∗ f satisﬁes Properties (i) and (ii) of Deﬁnition 3.2.1. Thus it remains to verify Properties (iii) and (iv).

Suppose the hypotheses of Property (iii). We see that

(Φ ∗ f)π(T1,...,Tn) X = Φσ(T1,...,Tn)f(σ, π) σ≤π X = Φ ((T ,...,T )| ) ··· Φ ((T ,...,T )| )f(σ| , π| ) ··· f(σ| , π| ) σ|V1 1 n V1 σ|Vm 1 n Vm V1 V1 Vm Vm σ≤π = (Φ ∗ f) ((T ,...,T )| ) ··· (Φ ∗ f) ((T ,...,T )| ), π|V1 1 n V1 π|Vm 1 n Vm using the bi-multiplicativity of Φ and the multiplicativity of f. 100 To see Property (iv) holds, note under the hypotheses of its initial case,

(Φ ∗ f)π(T1,...,Tn) X = Φσ(T1,...,Tn)f(σ, π) σ≤π X = Φσ| ((T1,...,Tθ−1,TθLΦ ((T ,...,T )| ),Tθ+1,...,Tn)|W )f(σ|V , π|V )f(σ|W , π|W ) W σ|V 1 n V σ≤π X = Φσ| ((T1,...,Tθ−1,TθLΦ ((T ,...,T )| )f(σ| ,π| ),Tθ+1,...,Tn)|W )f(σ|W , π|W ) W σ|V 1 n V V V σ≤π

= (Φ ∗ f)π| ((T1,...,Tθ−1,TθLΦ ((T ,...,T )| ),Tθ+1,...,Tn)|W ), W σ|V 1 n V again by the corresponding properties of Φ and f. The proof of the remaining three state- ments in Property (iv) is identical.

Corollary 3.4.4. The operator-valued bi-free cumulant function is bi-multiplicative.

3.4.3 Bi-moment and bi-cumulant functions.

Inspired by [22, Section 3.2], we deﬁne the formal classes of bi-moment and bi-cumulant functions and give an important relation between them. It follows readily that the operator- valued bi-free moment and cumulant functions on a B-B-non-commutative probability space are examples of these types of functions, respectively.

We begin with the following useful notation.

Notation 3.4.5. Let χ :[n] → {`, r}, π ∈ BN C(χ), and q ∈ [n]. We denote by χ|\q the restriction of χ to the set [n] \{q}. In addition, if q 6= n and χ(q) = χ(q + 1), we deﬁne

π|q=q+1 ∈ BN C(χ|\q) to be the bi-non-crossing partition which results from identifying q and q + 1 in π (i.e. if q and q + 1 are in the same block as π then π|q=q+1 is obtained from π by just removing q from the block in which q occurs, while if q and q + 1 are in diﬀerent blocks,

π|q=q+1 is obtained from π by merging the two blocks and then removing q).

Deﬁnition 3.4.6. Let (A, E, ε) be a B-B-non-commutative probability space and let

[ [ Φ: BNC(χ) × Aχ(1) × · · · × Aχ(n) → B n≥1 χ:[n]→{`,r} 101 be bi-multiplicative. We say that Φ is a bi-moment function if whenever χ :[n] → {`, r} is such that there exists a q ∈ {1, . . . , n − 1} with χ(q) = χ(q + 1), then

Φ1χ (T1,...,Tn) = Φ1 (T1,...,Tq−1,TqTq+1,Tq+2,...,Tn) χ|\q for all Tk ∈ Aχ(k). Similarly, we say that Φ is a bi-cumulant function if whenever χ :[n] → {`, r} and π ∈ BN C(χ) are such that there exists a q ∈ {1, . . . , n − 1} with χ(q) = χ(q + 1), then

X Φ1 (T1,...,Tq−1,TqTq+1,Tq+2,...,Tn) = Φ1χ (T1,...,Tn) + Φπ(T1,...,Tn) χ|\q π∈BN C(χ) |π|=2,qπq+1 for all Tk ∈ Aχ(k).

Notice that any operator-valued bi-free moment function E is a bi-moment function.

Before relating the notions of bi-moment and bi-cumulant functions, we note the following alternate formulations.

Lemma 3.4.7. Let (A, E, ε) be a B-B-non-commutative probability space and let

[ [ Φ: BNC(χ) × Aχ(1) × · · · × Aχ(n) → B n≥1 χ:[n]→{`,r} be bi-multiplicative. Then Φ is a bi-moment function if and only if whenever χ :[n] → {`, r} and π ∈ BN C(χ) are such that there exists a q ∈ {1, . . . , n − 1} with χ(q) = χ(q + 1) and q ∼π q + 1, then

Φπ(T1,...,Tn) = Φπ|q=q+1 (T1,...,Tq−1,TqTq+1,Tq+2,...Tn)

for all Tk ∈ Aχ(k). Similarly, Φ is a bi-cumulant function if and only if whenever χ :[n] → {`, r} is such that there exists a q ∈ {1, . . . , n − 1} with χ(q) = χ(q + 1), we have

X Φπ(T1,...,Tq−1,TqTq+1,Tq+2,...Tn) = Φσ(T1,...,Tn) σ∈BN C(χ) σ|q=q+1=π for all π ∈ BN C(χ|\q).

102 To establish the lemma, one uses bi-multiplicativity to reduce to the case of full partitions and then applies Deﬁnition 3.4.6.

Theorem 3.4.8. Let (A, E, ε) be a B-B-non-commutative probability space and let

[ [ Φ, Ψ: BNC(χ) × Aχ(1) × · · · × Aχ(n) → B n≥1 χ:[n]→{`,r} be related by the formulae

Φ = Ψ ∗ ζBNC, or equivalently, Ψ = Φ ∗ µBNC.

Then Φ is a bi-moment function if and only if Ψ is a bi-cumulant function.

Proof. To begin, note Φ is bi-multiplicative if and only if Ψ is bi-multiplicative by Theo- rem 3.4.3.

Suppose Ψ is a bi-cumulant function. If χ :[n] → {`, r} is such that there exists a q ∈ {1, . . . , n − 1} with χ(q) = χ(q + 1), then for all Tk ∈ Aχ(k)

X Φ1 (T1,...,Tq−1,TqTq+1,Tq+2,...,Tn) = Ψπ(T1,...,Tq−1,TqTq+1,Tq+2,...,Tn) χ|\q π∈BN C(χ|\q) X X = Ψσ(T1,...,Tn)

π∈BN C(χ|\q) σ∈BN C(χ) σ|q=q+1=π X = Ψσ(T1,...,Tn) σ∈BN C(χ)

= Φ1χ (T1,...,Tn).

Thus Φ is a bi-moment function.

For the other direction, suppose Φ is a bi-moment function. Let χ :[n] → {`, r}. We will proceed by induction on n. If n = 1, there is nothing to check. If n = 2, then

Ψ1 (T1T2) = Φ1 (T1T2) = Φ1 (T1,T2) = Ψ1 (T1,T2) + Ψ0 (T1,T2) χ|1=2 χ|1=2 χ χ χ as required.

103 Suppose that the formula from Deﬁnition 3.4.6 holds for n − 1. Then using the induction n o hypothesis and bi-multiplicativity of Ψ, we see for all π ∈ BN C(χ|\q) \ 1χ|\q that X Ψπ(T1,...,Tq−1,TqTq+1,Tq+2,...,Tn) = Ψσ(T1,...,Tn). σ∈BN C(χ) σ|q=q+1=π Hence

Ψ1 (T1,...,Tq−1,TqTq+1,Tq+2,...,Tn) χ|\q

= Φ1 (T1,...,Tq−1,TqTq+1,Tq+2,...,Tn) χ|\q X − Ψπ(T1,...,Tq−1,TqTq+1,Tq+2,...Tn)

π∈BN C(χ|\q) π6=1 χ|\q X X = Φ1χ (T1,...,Tn) − Ψσ(T1,...,Tn)

3.4.4 Vanishing of operator-valued bi-free cumulants.

The following demonstrates, like with classical and free cumulants, that operator-valued bi-free cumulants of order at least two vanish provided at least one entry is an element of B.

Proposition 3.4.10. Let (A, E, ε) be a B-B-non-commutative probability space, χ :[n] →

{`, r} with n ≥ 2, and Tk ∈ Aχ(k). If there exist q ∈ [n] and b ∈ B such that Tq = Lb if

χ(q) = ` or Tq = Rb if χ(q) = r, then

κ(T1,...,Tn) = 0.

Proof. We will proceed by induction on n. The base case can be readily established by computing directly the cumulants of order two. 104 For the inductive step, suppose the result holds for all χ : {1, . . . , k} → {`, r} with k < n.

Fix χ :[n] → {`, r} and Tk ∈ Aχ(k). Suppose that for some q ∈ [n] we have χ(q) = ` and

Tq = Lb with b ∈ B, as the argument for the right side is similar.

Let p = max {k ∈ [n] | χ(k) = `, k < q} .

The proof is now divided into two cases.

Case 1: p 6= −∞. In this case we notice that

κ1χ (T1,...,Tn) X = E1χ (T1,...,Tn) − κπ(T1,...,Tn) π∈BN C(χ) π6=1χ X = E1χ (T1,...,Tn) − κπ(T1,...,Tp−1,Tp,Tp+1,...,Tq−1,Lb,Tq+1,...,Tn) π∈BN C(χ) {q}∈π X = E1χ (T1,...,Tn) − κσ(T1,...,Tp−1,TpLb,Tp+1,...,Tq−1,Tq+1,...,Tn),

σ∈BN C(χ|\q)

by induction and Proposition 3.2.4. Since

E1χ (T1,...,Tn) = E(T1 ··· Tn)

= E(T1 ··· Tq−1LbTq+1 ··· Tn)

= E(T1 ··· TpLbTp+1 ··· Tq−1T1+1 ··· Tn) X = κσ(T1,...,Tp−1,TpLb,Tp+1,...,Tq−1,Tq+1,...,Tn),

σ∈BN C(χ|\q)

the proof is complete in this case.

105 Case 2: p = −∞. In this case, notice that

X κ1χ (T1,...,Tn) = E1χ (T1,...,Tn) − κπ(T1,...,Tn) π∈BN C(χ) π6=1χ X = E1χ (T1,...,Tn) − κπ(T1,...,Tq−1,Lb,Tq+1,...,Tn) π∈BN C(χ) {q}∈π X = E1χ (T1,...,Tn) − bκσ(T1,...,Tq−1,Tq+1,...,Tn),

σ∈BN C(χ|\q) by induction and Proposition 3.2.4. Since

E1χ (T1,...,Tn) = E(T1 ··· Tn)

= E(T1 ··· Tq−1LbTq+1 ··· Tn)

= E(LbT1 ··· Tq−1Tq+1 ··· Tn)

= bE(T1 ··· Tq−1Tq+1 ··· Tn) X = bκσ(T1,...,Tq−1,Tq+1,...,Tn),

σ∈BN C(χ|\q) the proof is complete in this case as well.

3.5 Universal moment polynomials for bi-free families with amalgamation.

In this section, we will generalize Corollary 2.2.3 to demonstrate that algebras will be bi- free with amalgamation over B if and only if certain moment expressions hold; our goal is to use the same technology to eventually show that bi-freeness with amalgamation can be characterised by the vanishing of mixed cumulants. To begin, we will need to extend the

deﬁnition of Eπ(T1,...,Tn) to certain diagrams in the style of Subsection 2.1.2 of Chapter 2.

3.5.1 Bi-freeness with amalgamation through universal moment polynomials.

lat Definition 3.5.1. Let χ :[n] → {`, r} and let ι :[n] → I. Let LRk (χ, ι) denote the closure lat of LRk(χ, ι) under lateral refinement. Observe that every diagram in LRk (χ, ι) still has k 106 strings reaching its top, as lateral refinements may only introduce cuts between ribs. We denote

lat [ lat LR (χ, ι) := LRk (χ, ι). k≥0 n ˚ o Definition 3.5.2. Let (Xι, Xι, pι) be B-B-bimodules with specified B-vector states, let ι∈I λι and ρι be as defined in Construction 3.1.5, and let X = ∗ι∈I Xι. Let χ :[n] → {`, r},

lat ι :[n] → I, D ∈ LR (χ, ι), and Tι ∈ Lχ(k)(Xι(k)). Define µk(Tk) = λι(k)(Tk) if χ(k) = ` and µk(Tk) = ρι(k)(Tk) if χ(k) = r. We define ED(µ1(T1), . . . , µn(Tn)) as follows: first, apply the same recursive process as in Definition 3.3.1 until every block of π has a spine reaching the top. If every block of D has a spine reaching the top, enumerate them from left to right according to their spines as V1,...,Vm with Vj = kj,1 < ··· < kj,qj , and set

E (µ (T ), . . . , µ (T )) = [(1−p )T ··· T 1 ]⊗· · ·⊗[(1−p )T ··· T 1 ]. D 1 1 n n ι(k1,1) k1,1 k1,q1 B ι(km,1) km,1 km,qm B

Lemma 3.5.3. With the notation as in Deﬁnition 3.5.2,   n X X  X |D|−|D0| µ1(T1) ··· µn(Tn)1B =  (−1)  ED(µ1(T1), . . . , µn(Tn)), lat  0  m=0 D∈LRm (χ,ι) D ∈LRm(χ,ι) 0 D ≥latD where |D| and |D0| denote the number of blocks of D and D0 respectively. In particular,   X  X  EL(X )(µ1(T1)µ2(T2) ··· µn(Tn)) =  µBNC(π, σ) Eπ(µ1(T1), . . . , µn(Tn)).   π∈BN C(χ) σ∈BN C(χ) π≤σ≤ι

Proof. To begin, note that the second claim follows from the ﬁrst by Deﬁnition 3.5.2 and by Proposition 2.4.1. To prove the main claim we will proceed by induction on the number of operators n. The case n = 1 is immediate.

For the inductive step, we will assume that χ(1) = ` as the proof in the case χ(1) = r

will follow by similar arguments. Let χ0 = χ|{2,...,n} and ι0 = ι|{2,...,n}. By induction, we obtain that   n−1 0 X X  X |D0|−|D0| µ2(T2) ··· µn(Tn)1B =  (−1)  ED0 (µ2(T2), . . . , µn(Tn)). m=0 lat  0  D0∈LRm (χ0,ι0) D0∈LRm(χ0,ι0) 0 D0≥latD0 107 The result will follow by applying λ1(T1) to each ED0 (µ2(T2), . . . , µn(Tn)), checking the correct terms appear, collecting the same terms, and verifying the coeﬃcients are correct.

lat Fix D0 ∈ LRm (χ0, ι0). We can write

ED0 (µ2(T2), . . . , µn(Tn)) = [(1 − pι(k1))S11B] ⊗ · · · ⊗ [(1 − pι(km))Sm1B]

for some operators Sp ∈ alg(λkp (L`(Xkp )), ρkp (Lr(Xkp ))). To demonstrate the correct terms appear, we divide the analysis into three cases.

Case 1: m = 0. In this case ED0 (µ2(T2), . . . , µn(Tn)) = b ∈ B. As such, we see that

λι(1)(T1)ED0 (µ2(T2), . . . , µn(Tn)) = E(T1)b ⊕ [(1 − pι(1))T1b].

If D1 is the LR-diagram obtained from D0 by placing a node shaded ι(1) at the top left and

D2 is the LR-diagram obtained from D0 by placing a node ι(1) at the top left and drawing a spine from this node to the top, then since

E(µ1(T1)Lb) = E(µ1(T1)Rb) = E(Rbµ1(T1)) = E(T1)b and

T1b = T1Rb1B = T1Lb1B one easily sees that

E(T1)b = ED1 (µ1(T1), µ2(T2), . . . , µn(Tn)) and

(1 − pι(1))T1b = ED2 (µ1(T1), µ2(T2), . . . , µn(Tn)).

Case 2: m 6= 0 and ι(1) 6= ι(k1). In this case, (1 − pι(k1))S11B is in a space orthogonal to ˚ Xι(1). Thus

λι(1)(T1)ED0 (µ2(T2), . . . , µn(Tn)) = [LE(T1)(1 − pι(k1))S11B] ⊗ · · · ⊗ [(1 − pι(km))Sm1B] ⊕ [(1 − pι(1))T11B] ⊗ [(1 − pι(k1))S11B] ⊗ · · · ⊗ [(1 − pι(km))Sm1B] . 108 If D1 is the LR-diagram obtained from D0 by placing a node shaded ι(1) at the top and D2 is the LR-diagram obtained from D0 by placing a node ι(1) at the top and drawing a spine from this node to the top, then since

LE(T1)(1 − pι(k1))S11B = (1 − pι(k1))LE(T1)S11B, one easily sees that

[LE(T1)(1 − pι(k1))S11B] ⊗ · · · ⊗ [(1 − pι(km))Sm1B] = ED1 (µ1(T1), µ2(T2), . . . , µn(Tn)) and

[(1 − pι(1))T11B] ⊗ [(1 − pι(k1))S11B] ⊗ · · · ⊗ [(1 − pι(km))Sm1B]

= ED2 (µ1(T1), µ2(T2), . . . , µn(Tn)).

Case 3: m 6= 0 and ι(1) = ι(k1). In this case, there is a spine in D that reaches the top ˚ and is the same shading as T1. Thus (1 − pι(k1))S11B ∈ Xι(1), so

λι(1)(T1)ED0 (µ2(T2), . . . , µn(Tn)) = [L (1 − pι(k ))S21B] ⊗ · · · ⊗ [(1 − pι(k ))Sm1B] pι(1)(T1(1−pι(1))S11B) 2 m ⊕ [(1 − pι(1))T1(1 − pι(1))S11B] ⊗ · · · ⊗ [(1 − pι(km))Sm1B] .

Expanding T1(1 − pι(1))S11B = T1S11B − T1E(S1), the above becomes

λι(1)(T1)ED0 (µ2(T2), . . . , µn(Tn)) = [LE(T1S1)(1 − pι(k2))S21B] ⊗ · · · ⊗ [(1 − pι(km))Sm1B] ⊕ (−1) [Lpι(1)(T1E(S1))(1 − pι(k2))S21B] ⊗ · · · ⊗ [(1 − pι(km))Sm1B] ⊕ [(1 − pι(1))T1S11B] ⊗ · · · ⊗ [(1 − pι(km))Sm1B] ⊕ (−1) [(1 − pι(1))T1E(S1)] ⊗ · · · ⊗ [(1 − pι(km))Sm1B] .

Let D1 be the LR-diagram obtained from D0 by placing a node shaded ι(1) at the top and terminating the left-most spine at this node, D2 be the LR-diagram obtained by laterally reﬁning D1 by cutting the spine attached to the top node directly beneath the top node, 109 D3 be the LR-diagram obtained from D0 by placing a node shaded ι(1) at the top and connecting this node to the left-most spine, and D4 be the LR-diagram obtained by laterally reﬁning D3 by cutting the spine attached to the top node directly beneath the top node. As in the previous case, we see (by applying Lemma 3.3.6 if m = 1) that

[LE(T1S1)(1 − pι(k2))S21B] ⊗ · · · ⊗ [(1 − pι(km))Sm1B] = ED1 (µ1(T1), µ2(T2), . . . , µn(Tn)) and

[(1 − pι(1))T1S11B] ⊗ · · · ⊗ [(1 − pι(km))Sm1B] = ED3 (µ1(T1), µ2(T2), . . . , µn(Tn)).

We will demonstrate that

[Lpι(1)(T1E(S1))(1 − pι(k2))S21B] ⊗ · · · ⊗ [(1 − pι(km))Sm1B] = ED2 (µ1(T1), µ2(T2), . . . , µn(Tn)) and forgo the similar proof that

[(1 − pι(1))T1E(S1)] ⊗ · · · ⊗ [(1 − pι(km))Sm1B] = ED4 (µ1(T1), µ2(T2), . . . , µn(Tn)).

Notice that

Lpι(1)(T1E(S1)) = Lp T R 1 = Lpι(1)(T11B)E(S1) = Lpι(1)(T11B)LE(S1), ι(1)( 1 E(S1) B) and so

Lpι(1)(T1E(S1))(1 − pι(k2))S21B = (1 − pι(k2))Lpι(1)(T11B)LE(S1)S21B.

Thus, unless m = 1, Lpι(1)(T11B) appears as it should in the deﬁnition of ED2 although the

E(S1) term may not be as it should. To obtain the desired result, we make the following corrections.

Recall that S1 corresponds to the left-most spine of D0 reaching the top. Let W ⊆

{2, . . . , n} be the set of k for which Tk appears in the expression for S1. Note that S1 will

0 0 be of the form C 0 C 0 ··· C 0 S , where each E(S ) is the moment of a disjoint E(S1) E(S2) E(Sp−1) p k

χ-interval Wk composed of blocks of D2 with the property that min≺χ (Wk) and max≺χ (Wk) Sp lie in the same block (C denotes either L or R, as appropriate). Observe that W = k=1 Wk. Therefore, by Proposition 3.1.4

0 00 00 E(S ) = E(C 0 C 0 ··· C 0 S ) = E(S ) ··· E(S ), 1 E(S1) E(S2) E(Sp−1) p 1 p 110 00 00 0 0 0 0 where the S1 ,...,Sp are S1,...,Sp up to reordering by ≺χ. Let W1,...,Wk be W1,...,Wk under this new ordering.

Through Lemma 3.3.6 in the case m = 1 and computations similar to the reverse of those used in cases 1 and 2 of Lemma 3.3.7 one can verify these terms can be moved into the correct positions.

Finally, it is clear that the coefficients of each ED(µ1(T1), . . . , µn(Tn)) are correct for each D ∈ LRlat(χ, ι). Alternatively, one can check the coefficients in the second claim by noting that the coefficients did not depend on the algebra B, setting B = C, and using Corollary 2.4.6.

Theorem 3.5.4. Let (A,EA, ε) be a B-B-non-commutative probability space, and suppose (ι) (ι) (ι) (ι) that (A` , Ar ) is a family of pairs of B-faces of A. Then (A` , Ar ) are bi-free ι∈I ι∈I (ι(k)) with amalgamation over B if and only if for all χ :[n] → {`, r}, ι :[n] → I, and Tk ∈ Aχ(k) , the following formula holds:   X  X  EA(T1 ··· Tn) =  µBNC(π, σ) Eπ(T1,...,Tn).   π∈BN C(χ) σ∈BN C(χ) π≤σ≤ι

(ι) (ι) Proof. If (A` , Ar ) are bi-free with amalgamation over B, then Lemma 3.5.3 implies ι∈I the desired formula holds.

Conversely, suppose that the formula holds. By Theorem 3.1.7 there exists a B-B- bimodule with a speciﬁed B-vector state (X , X˚, p) and a unital homomorphism θ : A → L(X ) such that

θ(ε(b1 ⊗ b2)) = Lb1 Rb2 , θ(A`) ⊆ L`(X ), θ(Ar) ⊆ Lr(X ), and EL(X )(θ(T )) = EA(T )

˚ ˚ for all b1, b2 ∈ B and T ∈ A. For each ι ∈ I, let (Xι, Xι, pι) be a copy of (X , X , p) and lι and rι be copies of θ : A → L(Xι). Since the formula holds for EL(X ) ◦ θ as well by Lemma 3.5.3, (ι) (ι) (A` , Ar ) are bi-free over B. ι∈I

111 3.6 Additivity of operator-valued bi-free cumulants.

3.6.1 Equivalence of bi-freeness with vanishing of mixed cumulants.

We now state the operator-valued analogue of the main result of Section 2.4 from Chapter 2, namely, that bi-freeness of families of pairs of B-faces is equivalent to the vanishing of their mixed cumulants.

Theorem 3.6.1. Let (A, E, ε) be a B-B-non-commutative probability space and suppose that (ι) (ι) (ι) (ι) (A` , Ar ) is a family of pairs of B-faces from A. Then (A` , Ar ) are bi-free ι∈I ι∈I with amalgamation over B if and only if for all χ :[n] → {`, r}, ι :[n] → I non-constant, (ι(k)) and Tk ∈ Aχ(k) , we have

κ1χ (T1,...,Tn) = 0.

(ι) (ι) Proof. Suppose (A` , Ar ) are bi-free over B. Fix a shading ι :[n] → I and let ι∈I χ :[n] → {`, r}. If T1,...,Tn are operators as above, Theorem 3.5.4 implies   X  X  E1χ (T1,...,Tn) =  µBNC(π, σ) Eπ (T1,...,Tn) .   π∈BN C(α) σ∈BN C(χ) π≤σ≤ι

Therefore X E1χ (T1,...,Tn) = κσ (T1,...,Tn) σ∈BN C(χ) σ≤ι by Deﬁnition 3.4.2. Using the above formula, we will proceed inductively to show that

κσ (T1,...,Tn) = 0 if σ ∈ BN C(χ) and σ ι. The base case where n = 1 holds vacuously. For the inductive case, suppose the result holds for every q < n. Suppose ι is not constant and note 1χ ι. Then

X X κσ (T1,...,Tn) = E1χ (T1,...,Tn) = κσ (T1,...,Tn) . σ∈BN C(χ) σ∈BN C(χ) σ≤ι

On the other hand, by induction and the bi-multiplicativity of κ, κσ (T1,...,Tn) = 0 provided

112 σ ∈ BN C(χ) \{1χ} and σ ι. Consequently,

X X κσ (T1,...,Tn) = κ1χ (T1,...,Tn) + κσ (T1,...,Tn) . σ∈BN C(χ) σ∈BN C(χ) σ≤ι

Combining these two equations gives κ1χ (T1,...,Tn) = 0, completing the inductive step.

Conversely, suppose all mixed cumulants vanish. Then we have

X E1χ (T1,...,Tn) = κσ (T1,...,Tn) σ∈BN C(χ) X = κσ (T1,...,Tn) σ∈BN C(χ) σ≤ι X X = Eπ (T1,...,Tn) µBNC(π, σ) σ∈BN C(χ) π∈BN C(χ) σ≤ι π≤σ   X  X  =  µBNC(π, σ) Eπ (T1,...,Tn) .   π∈BN C(χ) σ∈BN C(χ) π≤σ≤ι

(ι) (ι) Hence Theorem 3.5.4 implies (A` , Ar ) are bi-free over B. ι∈I

3.6.2 Moment and cumulant series.

In this section, we will begin the study of pairs of B-faces generated by operators.

Let (A, E, ε) be a B-B-non-commutative probability space and let (C,D) be a pair of B-faces such that

C = alg({Lb | b ∈ B} ∪ {zi}i∈I }) and D = alg({Rb | b ∈ B} ∪ {zj}j∈J }).

We desire to compute the joint distribution of (C,D) under E. To do so, it suﬃces to compute

E((C z C 0 ) ··· (C z C 0 )) = E (C 0 z C ,...,C 0 z C ) b1 α(1) b1 bn α(n) bn 1χα b1 α(1) b1 bn α(n) bn

0 0 where α :[n] → I t J, b , b , . . . , b , b ∈ B, and C = L , C 0 = L 0 if α(k) ∈ I and 1 1 n n bk bk bk bk

C = R , C 0 = R 0 if α(k) ∈ J. However, because E is bi-multiplicative it suﬃces to treat bk bk bk bk 113 0 the case where b = 1 and so C 0 = I; we may even assume b = 1. Similarly, to compute k bk n all possible cumulants, it suﬃces to compute

κ1χα (zα(1)Cb1 , zα(2)Cb2 , . . . , zα(n−1)Cbn−1 , zα(n)).

As such we make the following deﬁnition.

Deﬁnition 3.6.2. Let (A, E, ε) be a B-B-non-commutative probability space and let (C,D) be a pair of B-faces such that

C = alg({Lb | b ∈ B} ∪ {zi}i∈I }) and D = alg({Rb | b ∈ B} ∪ {zj}j∈J }).

The moment series of z = ((zi)i∈I , (zj)j∈J ) is the collection of maps

z n−1 µα : B → B | n ∈ N, α :[n] → I t J

given by

z µα(b1, . . . , bn−1) = E1χα (zα(1)Cb1 , zα(2)Cb2 , . . . , zα(n−1)Cbn−1 , zα(n)),

where Cbk = Lbk if α(k) ∈ I and Cbk = Rbk otherwise.

Similarly, the cumulant series of z is the collection of maps

z n−1 κα : B → B | n ∈ N, α :[n] → I t J

given by

z κα(b1, . . . , bn−1) = κ1χα (zα(1)Cb1 , zα(2)Cb2 , . . . , zα(n−1)Cbn−1 , zα(n)).

z z Note that if n = 1, we have µα = E(zα(1)) = κα.

Proposition 3.6.3. Let (A,E) be a B-B-non-commutative probability space, and for ι ∈ {0,00 } let {zι} ⊂ A and zι ⊂ A . If i i∈I ` j j∈J r Cι = alg {L : b ∈ B} ∪ {zι} and Dι = alg {R : b ∈ B} ∪ zι b i i∈I b j j∈J

are such that (C0,D0) and (C00,D00) are bi-free, then

z0+z00 z0 z00 κα = κα + κα .

Proof. This follows directly from Theorem 3.6.1. 114 3.7 Amalgamated versions of results from Chapter 2.

In this section we remark that several of our results from Chapter 2 have immediate or almost-immediate extensions to the operator-valued setting. We will eschew many of the details as they are for the most part restatements of the proofs in the scalar-valued case with much more tedious bookkeeping.

3.7.1 Amalgamated vaccine.

We ﬁrst turn our attention to the vaccine condition, which extends to the bi-free setting by replacing every instance of ϕ with E. Lemma 2.5.3 (that bi-free families exhibit vaccine) still holds in this situation, its proof being entirely combinatorial. One must take some care when reducing cumulants into products based on their blocks since B is potentially non- commutative; however, the key point of the argument is that if an interval is not connected to any nodes outside of it, its moment is multiplied into the product, and this still holds. All the terms corresponding to the isolated interval can be collected in one place in the correct

order, and then replaced by one of 0, L0, or R0.

The analogue of Lemma 2.5.4 requires a bit more care, however, essentially due to the fact that B is probably not algebraically closed, and even if it were, terms of the form

E ((z1 − b1) ··· (zn − bn)) (with zi ∈ A and bi ∈ ε(B ⊗ B)) do not directly reduce to polynomials with coeﬃcients in B, as the b’s cannot be assumed to commute past the z’s and so can’t be pulled out of the E without further argument. However, we do have the following Lemma which will suﬃce for our purposes:

Lemma 3.7.1. Suppose (A, E, ε) is a B-B-non-commutative probability space, with B a ˆ Banach algebra. Then for every χ :[n] → {`, r} and zi ∈ Aχ(i) there exist bi ∈ B so that, with b = ε(ˆb ⊗ 1) = L if χ(i) = ` and b = ε(1 ⊗ ˆb ) = R if χ(i) = r, we have i i ˆbi i i ˆbi

E ((z1 − b1) ··· (zn − bn)) = 0.

115 Proof. Let j = min≺χ ([n]). Notice that we can write

E ((z1 − b1) ··· (zn − bn)) = E ((z1 − b1) ··· (zj−1 − bj−1)zj(zj+1 − bj+1) ··· (zn − bn)) ˆ − bjE ((z1 − b1) ··· (zj−1 − bj−1)(zj+1 − bj+1) ··· (zn − bn)) .

Indeed, this is immediate if χ(j) = `, while if χ(j) = r we must be in the case j = n, so we can replace R by L and then pull ˆb out of the left. Now, if we take b = λ ∈ for i 6= j, ˆbn ˆbn n i C

we ﬁnd that bi commutes with every zk, and

n n−1 E ((z1 − λ) ··· (zj−1 − λ)(zj+1 − λ) ··· (zn − λ)) = (−λ) + O λ

becomes a polynomial in λ with coeﬃcients in B and leading term (−λ)n. In particular, for λ suﬃciently large it is invertible in B. Then once λ is large enough, we may take

ˆ bj = E ((z1 − λ) ··· (zj−1 − λ)zj(zj+1 − λ) ··· (zn − λ))

−1 · E ((z1 − λ) ··· (zj−1 − λ)(zj+1 − λ) ··· (zn − λ))

producing a solution to our equation.

Of course, we needed something slightly weaker than B being a Banach algebra: we only

require that monic polynomials with coeﬃcients in B have spectrum which is not all of C. With this lemma in hand, we can reprove Lemma 2.5.4 in the amalgamated setting; the only diﬀerence is that for each maximal χ-interval I we must choose a solution to an equation with |I| unknowns, rather than only one. It is of course important to note that if we have

Lb or Rb terms occurring in the expansion of this equation, they can be multiplied into adjacent z’s to still reduce the total number of variables in the moment being considered. We therefore have the following theorem:

(ι) (ι) Theorem 3.7.2. Suppose that B is a Banach algebra, and let (A` , Ar ) be a family ι∈I of pairs of B-faces in a B-B-non-commutative probability space (A, E, ε). Then the family has vaccine if and only if the pairs of B-faces are bi-free with amalgamation over B.

With vaccine established for the amalgamated setting, the results in Section 2.8 follow readily, using essentially the same proofs. In fact if one is more careful about bookkeeping, 116 and carefully studies the action of variables in their standard representation on a free product space, one can establish Theorems 2.8.3 and 2.8.4 when B is merely an algebra with no assumptions of closure; the details may be found in [8].

3.7.2 Amalgamated multiplicative convolution.

In the realm of multiplicative convolution things do not work out as nicely. We do still receive a direct analogue of Proposition 2.6.5: the combinatorial argument we employed goes through without diﬃculty. However we do not arrive at an equation like the one in Theorem 2.6.6: while it is true in the scalar setting that we can decompose a cumulant κπ(1)∪π(2) into a product κπ(1) · κπ(2) , this does not hold in the B-valued case: the terms corresponding to the second partition may be dispersed through the ﬁrst cumulant and impossible to collect.

117 CHAPTER 4

An investigation into regularity and free entropy.

In this section we collect some results dealing with the regularity of non-commutative random variables. We also show how some of the ideas from the study of free entropy, such as those in [29], and ideas from the study of free unitary Brownian motion coming from [4] may be used to gain further insight into bi-free probability.

4.1 Regularity results.

We begin by recalling several useful theorems from the literature, which we will use as a starting point for our results.

4.1.1 Useful results from the literature.

This ﬁrst result is paraphrased to suit our purposes.

Theorem 4.1.1 ([2, Theorem 2.9]). Suppose that µ is a non-atomic probability measure with algebraic Cauchy transform. Then µ has density f with respect to Lebesgue measure which

fails to exist at only ﬁnitely many points, and for some d > 0 and every a ∈ R satisﬁes

lim(x − a)1−df(x) < ∞. x→a

In particular, if 1 < p < (1 − d)−1, we have f ∈ Lp(R).

Theorem 4.1.2 ([1, Theorem 1]). Let (A, ϕ) be a non-commutative probability space. Let

x1, . . . , xn ∈ A be freely independent self-adjoint non-commutative random variables. Let

∗ X = X ∈ MN (C) ⊗ C hx1, . . . , xni ⊂ MN (C) ⊗ A 118 be a self-adjoint matrix polynomial. If the laws of x1, . . . , xn are algebraic, then so is the law of X.

Theorem 4.1.3 ([10, Theorem 3]). Let (M, τ) be a ﬁnite tracial von Neumann algebra, and

∗ y1, . . . , yn ∈ M. Suppose that Voiculescu’s free entropy dimension δ (y1, . . . , yn) = n. Then for any self-adjoint non-constant non-commutative polynomial P , the spectral measure of

y = P (y1, . . . , yn) has no atoms.

We will not deﬁne Voiculescu’s free entropy dimension, but instead state that the condi-

∗ tion in the above theorem is satisﬁed if χ(y1, . . . , yn) > −∞ or Φ (y1, . . . , yn) < ∞.

4.2 Algebraicity and ﬁnite entropy.

We the help of the following proposition, we are able to combine the above results to show that ﬁnite entropy is preserved under polynomial convolutions for suﬃciently smooth variables.

Proposition 4.2.1. Suppose that y = y∗ is a self-adjoint variable in a ﬁnite tracial von Neumann algebra (M, τ). Further suppose that the spectral distribution of y is Lebesgue

absolutely continuous, with density f. If f ∈ Lp(R) for some p > 1, then χ(y) > −∞.

Proof. Using interpolation and the fact that kfk1 = 1, we may assume that p < 2. From Theorem 1.1.20, we know that the free entropy of y is given by

3 1 χ(y) = log |s − t| dµ (s) dµ (t) + + log 2π, x y y 4 2 R2

so it suﬃces to bound the integral above. Let L(t) := 1(−4M,4M)(t) log |t|, where M = kyk

(so that the support of µy is contained in [−M,M]). Now for s ∈ R, we have Z Z f(s) f(t) log |s − t| dt = f(s) f(t)L(s − t) dt = f(s)(f ? L)(s).

119 We compute: Z Z

log |t − s| dµ(t) dµ(s) = f(s) f(t) log |s − t| dt ds x Z

= f(s)(f ? L)(s) ds

≤ kf · (f ? L)k1

≤ kfkp kf ? Lkq ,

1 1 where 1 = p + q , by H¨older’sinequality; note that q > 2 since p < 2. It suﬃces, now, to q q show that f ? L ∈ L (R); for this, we appeal to Young’s inequality. Indeed, if s = 2 > 1, 1 1 1 then 1 + q = p + s and so we have

kf ? Lkq ≤ kfkp kLks .

Yet kLks < ∞ for any 1 ≤ s < ∞, so we conclude |χ(x)| < ∞.

Corollary 4.2.2. Suppose that y1, . . . , yn are freely independent, self-adjoint, and algebraic, with χ(yi) > −∞. Then if P is a non-constant polynomial and y = P (y1, . . . , yn), we have χ(y) > −∞.

Proof. From Theorem 4.1.2, we know that y is algebraic. As y1, . . . , yn are free, we have

χ(y1, . . . , yn) = χ(y1)+...+χ(yn) > −∞, and so Theorem 4.1.3 informs us that the spectral measure of y has no atoms. Then Theorem 4.1.1 tells us that Proposition 4.2.1 applies, and we conclude that χ(y) > −∞.

It is tempting to believe that the above corollary is true far more generally, such as when

y1, . . . , yn is merely a system of variables with χ(y1, . . . , yn) > −∞. Indeed, one expects that polynomial convolution is an operation which should be well-behaved. However, we are not aware of a proof that works in this full generality.

4.2.1 Properties of spectral measures.

We will show in this section that certain strong regularity properties on the generators of an algebra do ensure weaker regularity of elements of that algebra. We begin by establishing 120 several useful lemmata which will allow us to gain some control over operators based on their spectral measures. The ﬁrst result may be found in [23, Section 4], though our narrower version of it essentially goes back to Kato [13]. We sketch a proof here for completeness.

Lemma 4.2.3. Let x = x∗ ∈ B(H) be a self-adjoint operator, and assume that the spectral measure of x is not Lebesgue absolutely continuous. Then there exists a sequence Tn of ﬁnite-rank operators which satisfy the following properties:

• 0 ≤ Tn ≤ 1;

• Tn → p weakly, where p is the spectral projection of x corresponding to the support of the Lebesgue-singular part of its spectral measure; and

•k [Tn, x]k1 → 0.

Proof. Replacing x by pxp, we may assume that x has singular spectral measure. Let us further assume that the spectrum of x is contained in [0, 1] (renormalizing and shifting x if

2 necessary) and that H = L ([0, 1], µx). Fix n > 0 and let E1,...,Ek be a disjoint collection of Borel subsets of [0, 1] such that

k k ! X 1 [ 1 diam(E ) < but µ E > 1 − . i n x i n i=1 i=1

Now let Qi be the rank 1 projection onto the function 1Ei , and Pn = Q1 + ... + Qk a rank

k projection. Plainly Pn → 1 weakly, by our choice of Ei. On the other hand,

k X [Pn, x] = [Qi, x]. i=1 1 Now [Qi, x] has rank at most two, while k[Qi, x]k ≤ diam(Ei) < n ; hence k X 2 k[P , x]k ≤ k[Q , x]k ≤ 2 diam(E ) ≤ → 0. n 1 i 1 i n i=1

The next two lemmata are similar to each other in spirit; each allows us to conclude that the vanishing of certain derivative-like quantities implies that a variable must be constant. First, though, we establish some convenient notation. 121 Notation 4.2.4. In what follows, given x, y ∈ A, we will denote y⊗x =: (x⊗y)σ, and extend this map linearly to all of A ⊗ A. Recall also the notation we established in Section 1.1.9: if X is an A-A-bimodule, ξ ∈ X, and x, y ∈ A, then (x ⊗ y)#ξ = x · ξ · y.

Lemma 4.2.5. Let (A, τ) be a non-commutative probability space with faithful tracial state

τ, generated by algebraically free self-adjoint elements y1, . . . , yn. Take y ∈ A a polynomial.

Let ∂i : A ⊗ A → A be the free diﬀerence quotients, as in Deﬁnition 1.1.22. Then y ∈

Alg (y1,..., yˆi, . . . , yn) if and only if

σ ∗ (∂iy) #y = 0.

Consequently, if the above equation holds for each i, y ∈ C.

Proof. One direction is immediate as Alg(y1,..., yˆi, . . . , yn) ⊆ ker ∂i.

Let Ni : A → A be the number operator associated to yi, the linear map which multiplies each monomial by its yi-degree. Observe that

(∂iy)#yi = Ni(y),

as each monomial m in y contributes P (a ⊗ b)#y = N (m). m=ayib i i

σ ∗ σ ∗ Suppose, then, that (∂iy) #y = 0, so yi(∂iy) #y = 0 as well. Let ϕλ : A → A be

the algebra homomorphism given by ϕλ(yi) = λyi, ϕλ(yj) = yj for j 6= i, which exists as y1, . . . , yn are algebraically free. We compute the following:

σ ∗ ∗ ∗ 0 = τ ◦ ϕλ(0) = τ ◦ ϕλ (yi(∂iy) #y ) = τ ◦ ϕλ (y (∂iy)#yi) = τ ◦ ϕλ (y Ni(y)) .

Now, suppose degyi (y) = d, and take x, z ∈ A so that x is yi-homogeneous of degree d, degyi (z) < d, and y = x + z. Then:

∗ ∗ ∗ ∗ 2d ∗ 2d−1 0 = τ ◦ ϕλ (y Ni(y)) = τ ◦ ϕλ(dx x) + τ ◦ ϕλ(z Ni(y) + x Ni(z)) = dλ τ(x x) + O λ .

Thus dλ2dτ(x∗x) = 0, and as τ is faithful, either x = 0 (in which case y = 0) or d = 0 (in

which case degyi (y) = 0). Either way, y ∈ Alg(y1,..., yî, . . . , yn). Repeated application of the above yields the final claim. 122 Lemma 4.2.6. Let y1, . . . , yn be algebraically free self-adjoint elements which generate a finite tracial von Neumann algebra (M, τ), and A be the algebra they generate. Suppose

∗ further that δ (y1, . . . , yn) = n. Once again, let ∂i : A ⊗ A → A be the free diﬀerence quotients. Suppose that N : A → A is a linear combination of number operators Ni, so for x ∈ A, n X N(x) = ai(∂ix)#yi. i=1 Let y = y∗ ∈ A be an eigenvector of N. Then if for some spectral projection p ∈ P(W ∗(y)) and for each 1 ≤ i ≤ n with ai 6= 0 we have

σ (∂iy) #(py) = 0, it follows that N(y) = 0.

Proof. As in the proof of Lemma 4.2.5, we compute:

n n ! X σ X 0 = aiτ (yi(∂iy) #(py)) = τ py ai(∂iy)#yi = τ (pyN(y)) = λτ (pyyp) . i=1 i=1 If λ 6= 0, it follows that pyyp = 0, hence py = 0. But then by Theorem 4.1.3 we have that y is constant and each Ni(y) = 0. In either case, N(y) = 0.

We now ready to state and prove the main result of this section.

Theorem 4.2.7. Let y1, . . . , yn be algebraically free self-adjoint elements which generate a ﬁnite tracial von Neumann algebra (M, τ), and A be the algebra they generate. Suppose further that y1, . . . , yn admit a dual system R1,...,Rn as in Deﬁnition 1.1.27. Take y =

∗ y ∈ A be a non-constant polynomial evaluated at (y1, . . . , yn). Then the spectral measure of y is not singular with respect to Lebesgue measure. Moreover, if there is N is in the positive linear span of the number operators Ni such that each Ni has a non-zero coeﬃcient and y is an eigenvector of N, then the spectral measure of y is Lebesgue absolutely continuous.

Proof. Assume to the contrary that the spectral measure of y is not absolutely continuous with respect to Lebesgue measure. By Lemma 4.2.3, we may choose 0 ≤ Tn ≤ 1 ﬁnite rank operators with Tn → p weakly and k[Tn, y]k1 → 0, where p is the spectral projection onto 123 the singular part of y. Let J : L2(M) → L2(M) be Tomita’s conjugation operator, deﬁned on x ∈ M by J(x) = x∗, and extended by continuity to L2(M).

Fix x ∈ M. Note that by deﬁnition of Rj,[Rj, yk] = ∂j(yk)#P1; since both [Rj, ·] and

∂j(·)#P1 are derivations, it follows that [Rj, y] = ∂j(y)#P1. We now compute as follows:

σ = |τ(((∂iy) #(yp))x)|

Here we used: the inequality kT wk1 ≤ kT k1 kwk∞, the trace property and commutation ∗ between Jx J and y, the equality [y, Ri] = −∂i(y)#P1, the fact that P1 is ﬁnite rank (so that we can pass to the limit Tn → p) and ﬁnally the equality τ(z) = hz1, 1i = Tr(zP1).

σ We conclude that (∂iy) #(yp) = 0 as τ is faithful and as x was arbitrary. Then if p = 1, Lemma 4.2.5 implies that y is constant, which is absurd; thus the spectral measure of y cannot be singular with respect to Lebesgue measure.

Further, if we have N as in the statement of the theorem, then Lemma 4.2.6 implies that N(y) = 0. It follows that for any non-zero monomial m in y, N(m) = 0. As m is an eigenvector of each Ni and each has a non-negative coeﬃcient in N, we learn that

Ni(m) = 0. Thus m has zero degree for each yi, and so m ∈ C. We conclude that y ∈ C, a contradiction.

4.3 Bi-free unitary Brownian motion.

Our aim in this section is to study multiplicative bi-free Brownian motion, as an analogue to the free unitary Brownian motion introduced by Biane [4]. Many related results in the free case were obtained in the context of a tracial von Neumann algebra, allowing the arguments 124 to be simpliﬁed; unfortunately that luxury is not available to us in the context of bi-free probability as we are not aware of an appropriate analogue of traciality. As the following example demonstrates, simply asking that the state on the non-commutative probability space be tracial is too restrictive.

(ι) (ι) Example 4.3.1. Suppose A` , Ar are bi-free pairs of faces in a non-commutative ι∈{ , } ( ) ( ) ( ) ( ) probability space (A, ϕ). Then we have for x ∈ A` , w ∈ Ar , y ∈ A` , and z ∈ Ar that

x w y x z y w z ϕ(xyzw) = ϕ(xw)ϕ(y)ϕ(z) + ϕ(x)ϕ(w)ϕ(yz) − ϕ(x)ϕ(w)ϕ(y)ϕ(z) while ϕ(wxyz) = ϕ(wx)ϕ(yz). Note that these two terms fail to be equal even when (x, w), (y, z) are a bi-free standard semicircular system with ϕ(wx) = ϕ(yz) = 1, as the left expression vanishes while the right equals 1.

4.3.1 Free Brownian motion.

We take some time to review the concept of free Brownian motion, which is the free analogue of the Gaussian process acting on a Hilbert space. A more complete description of free Brownian motion and free stochastic calculus may be found in [33].

Deﬁnition 4.3.2. A free Brownian motion in a non-commutative probability space (A, ϕ)

is a non-commutative stochastic process (S(t))t≥0 such that:

• the increments of S(t) are free: for 0 ≤ t1 < ··· < tk, the collection S(t2) −

S(t1),...,S(tk) − S(tk−1) are freely independent; and

• the process is stationary, with semicircular increments: for 0 ≤ s < t, S(t) − S(s) is semicircular with variance t − s.

Free Brownian motion can be modelled on a Fock space [33]. Indeed, suppose

2 M 2 ⊗n F L (R≥0) := CΩ ⊕ L (R≥0) . n≥1 125 ∗ Let ξt = 1[0,t], and deﬁne S(t) = l(ξt) + l (ξt). Then (S(t))t is a free Brownian motion.

Free unitary Brownian motion was initially introduced by Biane in [4] as a multiplicative analogue of the (additive) free Brownian motion above. It’s deﬁnition makes reference to a certain family of measures (νt)t≥0 supported on T, introduced by Bercovici and Voiculescu

in [3]. In particular, νt has the property that for t, s ≥ 0, νt νs = νt+s. We do not require the particular details of its introduction and so will eschew them.

Deﬁnition 4.3.3. A free unitary Brownian motion in a non-commutative probability space

(A, ϕ) is a non-commutative stochastic process (U(t))t≥0 such that:

• the (left) multiplicative increments of U(t) are free: for 0 ≤ t1 < ··· < tk, the in-

∗ ∗ ∗ crements given by U (t1)U(t2),U (t2)U(t3),...,U (tn−1)U(tn) are freely independent; and

∗ • the process is stationary with increments prescribed by ν·: the distribution of U (t)U(s)

depends only on s − t, and is in fact νs−t.

It was shown in [4] that if S(t) is a Fock space realization of a free additive Brownian motion and U(t) the solution to the free stochastic diﬀerential equation 1 dU(t) = iU(t) dS(t) − U(t) dt 2 with U(0) = 1, then U(t) is a free unitary Brownian motion. Moreover, the moments of a free unitary Brownian motion were computed: for n > 0,

n−1 X tk n ϕ(U(t)n) = (−1)k nk−1 e−nt/2. k! k + 1 k=0 A consequence is that free unitary Brownian motion converges in distribution as t → ∞ to

k a Haar unitary, i.e., a unitary u∞ with ϕ(u∞) = δk=0 for k ∈ Z. Another important results from [4] is the following bound: for some K > 0 and any t > 0, √ −t/2 U(t) − e ≤ K t.

We have already identiﬁed the bi-free analogue of Haar unitaries in Subsection 2.8.1, which we will use to motivate our approach to a bi-free version of Brownian motion: we want a 126 process which tends to the distribution of a Haar pair of unitaries, so that conjugating by the process asymptotically creates bi-freeness and can therefore be seen as a sort of liberation.

4.3.2 The free liberation derivation.

Suppose that A, B are algebraically free unital sub-algebras which generate a tracial non- commutative probability space (A, τ). In [29], Voiculescu deﬁned the derivation δA:B : A →

A ⊗ A to be a linear map satisfying the Leibniz rule such that δA:B(a) = a ⊗ 1 − 1 ⊗ a for

a ∈ A and δA:B(b) = 0 for b ∈ B. It was shown that A and B are freely independent if and

only if (τ ⊗ τ) ◦ δA:B ≡ 0. Moreover, the derivation δA:B relates to how the joint distribution of A and B changes as A is perturbed by unitary free Brownian motion.

Proposition 4.3.4 ([29, Proposition 5.6]). Let A, B be two unital ∗-subalgebras in (A, τ)

and let (U(t))t≥0 be a unitary free Brownian motion, which is freely independent of A ∨ B.

If aj ∈ A and bj ∈ B for 1 ≤ j ≤ n, then

∗ ∗ τ (U()a1U() b1 ··· U()anU() bn) n X = (τ ⊗ τ) δ a b ··· a b a b ··· a b 2 A:B k k n n 1 1 k−1 k−1 k=1 n !! X − bkak+1bk+1 ··· anbna1b1 ··· bk−1ak k=1 2 + τ(a1b1 ··· anbn) + O .

Important to the proof of the above proposition, and of use to us here also, is the following approximation result.

∗ Proposition 4.3.5 ([29, Proposition 1.4]). Let A be a W -subalgebra, (U(t))t a unitary free

Brownian motion, and S a (0, 1)-semicircular element in (M, τ) so that A and (U(t))t are

∗-free and A and S are also free. If aj ∈ A and αj ∈ {1, −1}, then we have

→ ! → ! Y Y t √ τ a U(t)αj = τ a 1 − + iα tS + O t2 , j j 2 j 1≤j≤n 1≤j≤n

where the products place the terms in order from left to right. 127 Although the proposition was stated in terms of a tracial W ∗-probability space, traciality was not needed in the proof.

4.3.3 A bi-free analogue to the liberation derivation.

For the remainder of this section, we will always be working in the context of a family of (ι) (ι) pairs of faces (A` , Ar ) generating a non-commutative probability space (A, ϕ). We ι∈I n (ι) o n (ι) o will further denote by A` and Ar the algebras generated by A` : ι ∈ I and Ar : ι ∈ I (ι) (ι) (ι) respectively, and by A the algebra generated by A` and Ar . Moreover, we assume that (i) (j) (i) (j) there are no algebraic relations between A and A other than [A` , Ar ] = 0 when i 6= j, (i) (i) and possibly [A` , Ar ] = 0. In particular, we want to ensure that given a product z1 ··· zn we can determine the χ-order of the variables.

Suppose χ :[n] → {`, r} and let 1 ≤ i, j ≤ n with i χ j. We denote [i, j]χ :=

{k : i χ k χ j} the χ-interval between i and j, and deﬁne analogously [i, j)χ,(i, j]χ, and

(i, j)χ. Likewise we deﬁne [i, ∞)χ := {k : 1 ≤ k ≤ n, i k} and analogously the other rays.

We will also change our conventions on the use of ι from earlier. For the remainder of this section, ι will always be an element of I, never a map; we will use to denote a map :[n] → I. ^ ^ Deﬁnition 4.3.6. Fix ι ∈ I. We deﬁne a map

(ι) W (j) : A → A ⊗ A A : j∈I\{ι} A ] as follows. Given χ :[n] → {`, r}, :[n] → I, and z ∈ A( (i)), i χ^(i) ^

(ι) W (j) (z1 ··· zn) A : j∈I\{ι} A ] X X c c c c = z[i,j]χ ⊗ z[i,j]χ − z[i,j)χ ⊗ z[i,j)χ − z(i,j]χ ⊗ z(i,j]χ + z(i,j)χ ⊗ z(i,j)χ . i∈ −1(ι) j∈ −1(ι) ^ i^χj We now extend this deﬁnition by linearity to all of A. When context makes our intent clear,

we will sometimes write ι for (ι) W (j) . A : j∈I\{ι} A ] ] The subscript A : B is meant to mimic that in the free situation, and the basic properties 128 present there still hold: B ⊂ ker A:B and for a ∈ A, A:B(a) = 1 ⊗ a − a ⊗ 1. However, ] ] A:B is not a derivation, even when restricted to the left or right faces of A and B. ] Lemma 4.3.7. ι is well-defined. Moreover, the only terms which do not cancel in the sum ] defining ι are those in which no -monochromatic χ-interval is split across a tensor sign. ] ^ Proof. Our assumptions about the lack of algebraic relations in A mean that the only ambiguity in writing a product z1 ··· zn comes from grouping or failing to group adjacent terms, and commuting left and right terms; the latter has no impact on ι because it does not ] change the χ-ordering of the variables. Notice that if i ≺χ i+ are consecutive under the χ-ordering and both contribute to the sum, then all intervals with i as an open left endpoint are intervals with i+ as a closed left endpoint and have opposite sign in their contributions to the two terms; likewise, all intervals with i as a closed right endpoint are intervals with i+ as an open right endpoint and again cancel. Hence the value of ι does not change if a product is written differently, and the only terms which do not cancel] are those with the tensor sign falling between two -monochromatic χ-intervals (or one such interval and the edge of the product), exactly one^ of which is ι-coloured.

In essence, ι acts by adding one term for each χ-interval with endpoints either before or after terms coming] from A(ι), consisting of the product of the terms not in that interval tensored with the product of the terms in the interval. The sign is chosen so that if the divi- sion comes before both chosen nodes or after both chosen nodes the term counts negatively,

and otherwise counts positively. Notice that when i = j, the terms corresponding to [i, i)χ

and (i, i)χ cancel and only one term contributing −z1 ··· zn ⊗ 1 survives.

The liberation gradient δA(ι):B can be expressed in a similar manner:

X δA:B(z1 ··· zn) = −z(−∞,i) ⊗ z(−∞,i)c + z(−∞,i] ⊗ z(−∞,i]c . i∈ −1(ι) ^ Example 4.3.8. Let χ :[n] → {`, r} and :[n] → I be as in Example 2.5.2. Then ^

129 (z1 ··· z10) is a sum of the following eight terms: ]

1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7 7 8 8 8 8 9 9 9 9 10 10 10 10 −z1 ··· z10 ⊗ 1 z1 ··· z5z8z9z10 ⊗ z6z7 −z1 ··· z5 ⊗ z6 ··· z10 z1z2z3z5 ⊗ z4z6 ··· z10 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7 7 8 8 8 8 9 9 9 9 10 10 10 10 z1 ··· z7 ⊗ z8z9z10 −z1z2z3z5z6z7 ⊗ z4z8z9z10 z1z2z3z5 ··· z10 ⊗ z4 −z1 ··· z10 ⊗ 1

(1) (1) Theorem 4.3.9. Let the notation be as above, and suppose I = {1, 2}. Then (A` , Ar ) (2) (2) and (A` , Ar ) are bi-free if and only if (ϕ ⊗ ϕ) ◦ 1 ≡ 0. ]

Proof. Suppose ﬁrst that bi-freeness holds. Note that for λ ∈ C, 1(λ) = 0, so it suf- ﬁces to check the condition on products z ··· z with z ∈ A( (i)) and] each maximal - 1 n i χ^(i) monochromatic χ-interval centred, since an arbitrary term may be written as a sum^ of such terms. However, by Lemma 4.3.7 we know that each term in 1(z1 ··· zn) is a tensor product with zero or more centred χ-intervals occurring on each side] of the tensor. The vaccine condition from bi-freeness then tells us that ϕ ⊗ ϕ of such a term is 0, and so

(ϕ ⊗ ϕ) ◦ 1(z1 ··· zn) = 0. ] Now, suppose that bi-freeness fails, and let z1, . . . , zn be an example of the failure of vaccine with a minimum number of terms. Then the only terms which possibly fail to vanish under ϕ ⊗ ϕ from 1 are those of the form z1 ··· zn ⊗ 1 or 1 ⊗ z1 ··· zn (the rest being ones to which vaccine] should apply, which are of shorter length and so not counterexamples by minimality). The term z1 ··· zn ⊗ 1 occurs once per 1-coloured χ-interval with negative sign, while 1 ⊗ z1 ··· zn occurs once with positive sign if the χ-ﬁrst and χ-last variables are both in A(1), and not at all otherwise; let k be the number of 1-coloured χ-intervals, and d = 1 if the χ-ﬁrst and χ-last variables are in A(1), with d = 0 otherwise. Hence

ϕ ⊗ ϕ( 1(z1 ··· zn)) = (d − k)ϕ(z1 ··· zn) 6= 0 unless k = d; but if k = d either there is one 1-interval] which is [n], or there are no 1-intervals; in either case, is constant and so 130 ^ z1 ··· zn cannot actually be a counterexample of vaccine.

4.3.4 Bi-free unitary Brownian motion.

We are now ready to introduce a notion of bi-free unitary Brownian motion.

Deﬁnition 4.3.10. A pair of free stochastic processes (U`(t),Ur(t))t≥0 is a bi-freely liberating unitary Brownian motion (or bi-ﬂu Brownian motion) if:

• the multiplicative increments are bi-free: if 0 ≤ t1 < ··· < tn, then the family of pairs

∗ ∗ n−1 of faces ((U` (tι)U`(tι+1),Ur(tι+1)Ur (tι))ι=1 is bi-free;

∗ • (U`(t))t≥0 and (Ur (t))t≥0 are each free unitary Brownian motions, and for all t > 0 the ∗ ∗-distribution of the pair (U`(t),Ur(t)) matches that of (U`(t),U` (t)); and

∗ ∗ • the distribution is stationary: the moments of (U` (s)U`(t),Ur(t)Ur (s)) depend only on t − s.

We have qualiﬁed this as a liberating unitary Brownian motion because it asymptotically introduces bi-free independence without modifying the distributions of the faces being lib- erated; we reserve the possibility of using “bi-free unitary Brownian motion” more broadly, such as for processes with diﬀerent covariance between the left and right faces.

We will show that once again a bi-ﬂu Brownian motion may be realized from an additive Brownian motion.

Lemma 4.3.11. Suppose that (S`(t))t≥0 is a free Brownian motion in a tracial von Neumann algebra (M, τ), and J : L2(M) → L2(M) is the Tomita operator deﬁned on M by J(x) = x∗

2 0 and extended continuously to L (M). Let Sr(t) = JS`(t)J ∈ M . Then if (U`(t),Ur(t)) are solutions to the stochastic diﬀerential equations

1 1 dU (t) = iU (t) dS (t) − U (t) dt and dU (t) = −iU (t) dS (t) − U (t) dt, ` ` ` 2 ` r r r 2 r with initial conditions U`(0) = 1 = Ur(0), the pair (U`(t),Ur(t)) is a bi-ﬂu Brownian motion.

Moreover, (U`(t),Ur(t)) converges in distribution as t → ∞ to a Haar pair of unitaries. 131 Proof. We ﬁnd immediately that U`(t) is a unitary free Brownian motion. Note that inte-

2 grating a stochastic process ωt]dXt comes down to ﬁnding a limit in L (A) of approxima- P P tions of the form θtk (xtk − xtk−1)φtk , where θtk ⊗ φtk approximates ωt. It follows that ∗ ∗ d(JXt J) = J(dXt) J, and in particular, JdSr(t)J = dS`(t). Conjugating the equation for dUr(t) above, we ﬁnd

1 1 d(JU (t)J) = i (JU (t)J) JdS (t)J − (JU (t)J) dt = i (JU (t)J) dS (t) − (JU (t)J) dt. r r r 2 r r ` 2 r

Thus JUr(t)J satisﬁes the same diﬀerential equation as U`, whence the two are equal. We

conclude that Ur(t) corresponds to right multiplication in the standard representation on

2 ∗ L (M) by U` (t). The remaining properties of bi-ﬂu Brownian motion now follow readily from

the free properties possessed by (U`(t))t≥0, using, essentially, the techniques of Theorem 2.8.4.

We find that conjugating by bi-flu Brownian motion leads to bi-freeness as t → ∞, much like in the free case, and this allows to think of this as a sort of bi-free liberation process. A strange consequence is the following: suppose that X,Y ∈ L∞(Ω, µ) ⊂ A are classical random variables, and ((U`(t),Ur(t))t≥0 a bi-flu Brownian motion in A, bi-free from

∗ (X,Y ). Then X commutes in distribution with Y , Ur(t), and Ur (t), so in particular, X ∗ and Ur(t)YUr (t) become independent as t → ∞ while always generating a commutative probability space. One ﬁnds that

∗ ∗ ϕ(f(X)Ur(t)g(Y )Ur (t)) = ϕ(f(X)g(Y ))ϕ(Ur(t))ϕ(Ur (t))

∗ + ϕ(f(X))ϕ(g(Y )) (1 − ϕ(Ur(t))ϕ(Ur (t))) = ϕ(f(X)g(Y ))e−t + ϕ(f(X))ϕ(g(Y )) 1 − e−t .

We will demonstrate a connection between liberation and the map , but ﬁrst we need a bi-free version of Proposition 4.3.5. ]

Lemma 4.3.12. Suppose (A`, Ar) is a pair of faces in A and (U`(t),Ur(t)) is a bi-ﬂu Brow-

nian motion, bi-free from (A`, Ar). Suppose further that (S`,Sr) is a pair of semicircular

variables with covariance matrix containing a 1 in every entry, also bi-free from (A`, Ar). 132 Let χ :[n] → {`, r}, and for 1 ≤ j ≤ n, take aj ∈ Aχj and αj ∈ {1, 0, −1}. Deﬁne

ψ :[n] → {1, −1} by ψ(j) = αj if χ(j) = `, and ψ(j) = −αj otherwise. Then we have

→ ! → ! Y Y t √ ϕ a U (t)αj = ϕ a 1 − |α | + iψ(j) tS + O t2 , j χ(j) j j 2 χ(j) 1≤j≤n 1≤j≤n

Essentially, this lemma tells us that the pair (U`(t),Ur(t)) behaves in ∗-distribution to √ √ t t order t the same as the pair 1 − 2 + i tS`, 1 − 2 − i tSr .

Proof. We proceed along the same lines as in the proof of Proposition 4.3.5. Let I =

{j : αj 6= 0}, and write m := |I|. Since the ∗-distribution of (U`(t),Ur(t)) is the same as that

∗ of (U`(t),U` (t)), one can check that for any sequence j1 < . . . < jk of terms in I,

αj1 −t/2 αjk −t/2 2 ϕ (Uχ(j1)(t) − e ) ··· (Uχ(jk)(t) − e ) = −δk=2ψ(j1)ψ(j2)t + O t .

This follows from the fact that the same is true in the free case, which was used in the original proof of Proposition 4.3.5 (cf. [29]).

αj αj −t/2 −t/2 Now for each j ∈ I, we rewrite Uχ(j)(t) as Uχ(j)(t) − e + e , and expand the √ αj −t/2 product. As we have the estimate Uχ(j)(t) − e ≤ K t, we ﬁnd that only terms where at most three of these are chosen will contribute more than O (t2). But by the above argument, terms with one or three such diﬀerences are O (t2) under ϕ; then only terms

αj −t/2 which contribute are those where precisely zero or two Uχ(j)(t) − e terms are chosen. Hence, if we abbreviate  

 X αp −t/2 αq −t/2  Z(t) :=  ϕ a1 ··· ap(U − e )ap+1 ··· aq(U − e )aq+1 ··· an  ,  χ(p) χ(q)  1χp≺χqχn p,q∈I

133 we have

→ ! Y αj −mt/2 2 −(m−2)t/2 ϕ ajUχ(j)(t) = ϕ(a1 ··· an)e + O t − e Z(t) 1≤j≤n

−mt/2 2 = ϕ(a1 ··· an)e + O t  

−(m−2)t/2  X  c − te  ϕ(a(p,q]χ )ϕ(a(p,q] )ψ(p)ψ(q)  χ  1χp≺χqχn p,q∈I t = ϕ(a ··· a ) 1 − m + O t2 1 n 2    X  c − t  ϕ(a(p,q]χ )ϕ(a(p,q] )ψ(p)ψ(q) .  χ  1χp≺χqχn p,q∈I

Here the second equality may require some justiﬁcation. One can verify that it is correct by considering the expansion in terms of cumulants; the terms corresponding to partitions with blocks of mixed colour or partitions that do not connect the U terms both vanish, and we are left with all the bi-non crossing partitions which have the two joined. Summing over these, in turn, produces the product of the two moments claimed.

Next we turn our attention to the right hand side of the equation. Notice that the pair

(S`,Sr) has the same distribution as (−S`, −Sr) while both are bi-free from (A`, Ar), so √ √ replacing t by − t does not change the value and thus we are in fact working with a √ power series in t rather than t. Since the constant term is clearly correct, we need only establish that the t term agrees. Contributions to the linear term come either from selecting

t t a single 2 in the product (together these contribute −m 2 ϕ(a1 ··· an)) or from selecting a pair indices to include the semicircular terms from. But now √ √ ϕ(a1 ··· ap(iψ(p) t)Sχ(p)ap+1 ··· aq(iψ(q) t)Sχ(q)aq+1 ··· an)

c = −tψ(p)ψ(q)ϕ(a(p,q]χ )ϕ(a(p,q]χ ).

Summing over the terms from which semi-circular elements may be selected, which is to say those with indices coming from I, we see the two sides of the claimed equation agree at order t, also. 134 (ι) (ι) Theorem 4.3.13. Suppose (A` , Ar )ι∈{ , } are algebraically-free pairs of faces in a non- commutative probability space (A, ϕ), bi-free from the bi-ﬂu Brownian motion (U`(t),Ur(t)).

Given χ :[n], :[n] → { , }, and xi ∈ Aχ(i), set ^   xi if (i) = zi(t) = ∗ ^  Uχ(i)(t)xiUχ(i)(t) if (i) = . ^ Then we have the following estimate:

2 ϕ(z1(t) ··· zn(t)) = ϕ(x1 ··· xn) + tϕ ⊗ ϕ ( (x1 ··· xn)) + O t . ] √ ±1 t ±1 Proof. We ﬁrst apply Lemma 4.3.12 to replace U`(t) by 1 − 2 ± i tS` and Ur(t) by √ t 1 − 2 ∓ i tSr, for some (S`,Sr) bi-free from (A`, Ar) as in Lemma 4.3.12. Again, as the

distribution of (S`,Sr) matches that of (−S`, −Sr), we ﬁnd that we are dealing with a power series in t; further, it is evident that the constant term is correct. We therefore consider contributions to the linear term.

However, note that these precisely correspond to the terms in the deﬁnition of . Indeed, ] we notice that when i ≺χ j with (i) = (j) = , selecting the S terms on either side of xi ^ ^ and xj contribute a total of

c c c c t ϕ(x[i,j]χ )ϕ(x[i,j]χ ) − ϕ(x[i,j)χ )ϕ(x[i,j)χ ) − ϕ(x(i,j]χ )ϕ(x(i,j]χ ) + ϕ(x(i,j)χ )ϕ(x(i,j)χ ) .

The signs occur because the signs of S’s χ-before their respective elements, or χ-after, always match. This accounts for all the contributions coming from selecting two semicircular vari-

t ables when expanding the product; what’s left are the terms corresponding to selecting a − 2

term, so each xi coming from winds up contributing −tϕ(x1 ··· xn) in total. Yet this precisely matches the contribution to corresponding to selecting the empty terms with i = j. ] We conclude that the linear term in ϕ(z1(t) ··· zn(t)) is precisely tϕ ⊗ ϕ ( (x1 ··· xn)). ] In [29], Voiculescu used the free liberation process to deﬁne the liberation gradient and a mutual non-microstates free entropy. Thus our approach here may be seen as taking the ﬁrst steps towards a non-microstates theory of bi-free entropy.

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