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Free Probability Notes Free probability and combinatorics Preliminary version Michael Anshelevich c 2012 July 17, 2018 Preface These notes were used in a topics course Free probability and combinatorics taught at Texas A&M Uni- versity in the fall of 2012. The course was based on Nica and Speicher’s textbook [NS06], however there were sufficient differences to warrant separate notes. I tried to minimize the list of references, but many more are included in the body of the text. Thanks to the students in the course, and to March Boedihardjo, for numerous comments and corrections; further corrections or comments are welcome. Eventually I will probably replace hand-drawn figures by typeset ones. Contents 1 Noncommutative probability spaces and distributions. 3 1.1 Non-commutative probability spaces. 3 1.2 Distributions. Weak law of large numbers. 6 2 Functional analysis background. 9 2.1 C∗- and von Neumann algebras. Spectral theorem. 9 2.2 Gelfand-Naimark-Segal construction. 11 2.3 Other topics. 14 3 Free independence. 19 3.1 Independence. Free independence. 19 3.2 Free products. 22 4 Set partition lattices. 29 4.1 All, non-crossing, interval partitions. Enumeration. 29 4.2 Incidence algebra. Multiplicative functions. 33 5 Free cumulant machinery. 38 5.1 Free cumulants. 38 5.2 Free independence and free cumulants. 45 5.3 Free convolution, distributions, and limit theorems. 49 1 6 Lattice paths and Fock space models. 54 6.1 Dyck paths and the full Fock space. 54 6.2 Łukasiewicz paths and Voiculescu’s operator model. 62 6.3 Motzkin paths and Schurmann’s¨ operator model. Freely infinitely divisible distributions. 63 6.4 Motzkin paths and orthogonal polynomials. 69 6.5 q-deformed free probability. 74 7 Free Levy´ processes. 77 8 Free multiplicative convolution and the R-transform. 81 9 Belinschi-Nica evolution. 85 10 ∗-distribution of a non-self-adjoint element 93 11 Combinatorics of random matrices. 97 11.1 Gaussian random matrices. 97 11.2 Map enumeration. 104 11.3 Partitions and permutations. 108 11.4 Asymptotic free independence for Gaussian random matrices. 112 11.5 Asymptotic free independence for unitary random matrices. 115 12 Operator-valued free probability. 121 13 A very brief survey of analytic results. 128 13.1 Complex analytic methods. 128 13.2 Free entropy. 132 13.3 Operator algebras. 136 2 Chapter 1 Noncommutative probability spaces and distributions. See Lectures 1, 4, 8 of [NS06]. 1.1 Non-commutative probability spaces. Definition 1.1. An (algebraic)(non-commutative) probability space is a pair (A;'), where A is a unital ∗-algebra and ' is a state on A. That is: •A is an algebra over C, with operations za + wb, ab for z; w 2 C, a; b 2 A. Unital: 1 2 A. • ∗ is an anti-linear involution, (za)∗ = za∗, (ab)∗ = b∗a∗. • ' : A! C is a linear functional. Self-adjoint: ' [a∗] = ' [a]. Unital: ' [1] = 1. Positive: ' [a∗a] ≥ 0. Definition 1.2. a 2 A is symmetric if a = a∗. A symmetric a 2 (A;') is a (n.c.) random variable. Examples of commutative probability spaces. Example 1.3. Let (X; Σ;P ) be a measure space, P a probability measure. Take Z 1 ∗ A = L (X; P ); f = f; E(f) = f dP: X Then E (usually called the expectation) is a state, in particular positive and unital. Thus (A; E) is a (commutative) probability space. Note: f = f ∗ means f real-valued. 3 A related construction is \ A = L∞−(X; P ) = Lp(X; P ); p≥1 the space of complex-valued random variables all of whose moments are finite. Example 1.4. Let X be a compact topological space (e.g. X = [0; 1]), and µ a Borel probability measure X on . Then for Z A = C(X);' [f] = f dµ, X (A;') is again a commutative probability space. Example 1.5. Let A = C[x] (polynomials in x with complex coefficients), x∗ = x. Then any state ' on C[x] gives a commutative probability space. Does such a state always come from a measure? Examples of non-commutative probability spaces. Example 1.6. Let x1; x2; : : : ; xd be non-commuting indeterminates. Let A = Chx1; x2; : : : ; xdi = Chxi be polynomials in d non-commuting variables, with the involution ∗ ∗ xi = xi; (xu(1)xu(2) : : : xu(n)) = xu(n) : : : xu(2)xu(1): Then any state ' on Chxi gives a non-commutative probability space. These do not come from measures. d One example: for z = (z1; z2; : : : ; zd) 2 R , δz(f(x1; x2; : : : ; xd)) = f(z1; z2; : : : ; zd) is a state (check!). Other examples? ∗ Example 1.7. A = Mn(C), the n × n matrices over C, with the involution (A )i;j = Aji and n 1 X 1 ' [A] = A = Tr(A) = tr(A); N ii N i=1 the normalized trace of A. Note that indeed, tr(A∗) = tr(A) and tr(A∗A) ≥ 0. Definition 1.8. Let ' be a state on A. a. ' is tracial, or a trace, if for any a; b 2 A, ' [ab] = ' [ba] : Note that A is, in general, not commutative. 4 b. ' is faithful if ' [a∗a] = 0 only if a = 0. Example 1.9. For a probability space (X; Σ;P ), let ∞− ∞− A = Mn(C) ⊗ L (X; P ) ' Mn(L (X; P )): These are random matrices = matrix-valued random variables = matrices with random entries. Take Z ' [A] = (tr ⊗ E)(A) = tr(A) dP: X Example 1.10. Let H be a Hilbert space, and A a ∗-subalgebra of B(H), the algebra of bounded linear operators on H. a∗ is the adjoint operator to a. If ξ 2 H is a unit vector, then ' [a] = haξ; ξi is a state. Why unital: ' [1] = hξ; ξi = kξk2 = 1: Why self-adjoint: ' [a∗] = ha∗ξ; ξi = hξ; aξi = haξ; ξi: Why positive: ' [a∗a] = ha∗aξ; ξi = haξ; aξi = kaξk2 ≥ 0: Typically not tracial or faithful. n Example 1.11 (Group algebra). Let Γ be a discrete group (finite, Z , Fn, etc.). C[Γ] = functions Γ ! C of finite support = finite linear combinations of elements of Γ with C coefficients: X (f : x 7! f(x)) $ f(x)x: x2Γ This is a vector space. It is an algebra with multiplication ! ! X X X f(x)x g(y)y = (f(x)g(y))z; x2Γ y2Γ z2Γ z=xy in other words X X (fg)(z) = f(x)g(y) = f(x)g(x−1z); z=xy x2Γ the convolution multiplication. The involution is f ∗(x) = f(x−1): 5 Check that indeed, (fg)∗ = g∗f ∗. So C[Γ] is a unital ∗-algebra, with the unit δe, where for x 2 Γ ( 1; y = x; δx(y) = 0; y 6= x and e is the unit of Γ. Moreover, define τ [f] = f(e): Exercise 1.12. Prove that τ is a faithful, tracial state, called the von Neumann trace. Later: other versions of group algebras. 1.2 Distributions. Weak law of large numbers. Definition 1.13. Let (A;') be an n.c. probability space. a. Let a 2 (A;') be symmetric. The distribution of a is the linear functional 'a : C[x] ! C;'a [p(x)] = ' [p(a)] : Note that 'a is a state on C[x]. The sequence of numbers n n fmn['a] = 'a [x ] = ' [a ] ; n = 0; 1; 2;:::g are the moments of a. In particular m0['a] = 1 and m1['a] = ' [a] is the mean of a. b. More generally, let a1; a2; : : : ; ad 2 (A;') be symmetric. Their joint distribution is the state 'a1;:::;ad : Chx1; : : : ; xdi ! C;'a1;:::;ad [p(x1; : : : ; xd)] = ' [p(a1; : : : ; ad)] : The numbers ' au(1)au(2) : : : au(n) : n ≥ 0; 1 ≤ u(i) ≤ d are the joint moments of a1; : : : ; ad. Denote by D(d) the space of all joint distributions of d-tuples of symmetric random variables, which is the space of all states on Chx1; : : : ; xdi. 6 c. We say that (N) (N) (a1 ; : : : ; ad ) ! (a1; : : : ; ad) in moments (or, for d > 1, in distribution) if for each p, ' (N) (N) [p] ! '(a1;:::;ad) [p] (a1 ;:::;ad ) as N ! 1. Remark 1.14. Each µ 2 D(d) can be realized as a distribution of a d-tuple of random variables, namely (x1; x2; : : : ; xd) ⊂ (Chxi; µ). Definition 1.15. a1; a2; : : : ; ad 2 (A;') are singleton independent if ' au(1)au(2) : : : au(n) = 0 whenever all ai are centered (that is, ' [ai] = 0) and some index in ~u appears only once. Proposition 1.16 (Weak law of large numbers). Suppose fan : n 2 Ng ⊂ (A;') are singleton indepen- dent, identically distributed (that is, all 'ai are the same), and uniformly bounded, in the sense that for a fixed C and all ~u, n ' au(1)au(2) : : : au(n) ≤ C : Denote 1 s = (a + a + ::: + a ): n n 1 2 n Then sn ! ' [a1] in moments (sn converges to the mean, a scalar). Proof. Note first that 1 (a − ' [a ]) + ::: + (a − ' [a ]) = s − ' [a ] : n 1 1 n n n 1 So without loss of generality, may assume ' [a1] = 0, and we need to show that sn ! 0 in moments. n n 1 1 X X ' sk = ' (a + ::: + a )k = ::: ' a a : : : a : n nk 1 n nk u(1) u(2) u(k) u(1)=1 u(k)=1 How many non-zero terms? Denote B(k) the number of partitions of k points into disjoint non-empty subsets. Don’t need the exact value, but see Chapter 4. For each partition π = (B1;B2;:::;Br);B1 [ B2 [ :::Br = f1; : : : ; kg ;Bi \ Bj = ; for i 6= j: How many k-tuples (u(1); u(2); : : : ; u(k)) such that u(i) = u(j) , i ∼π j ( i.e.
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