Lecture Notes on “Completely Positive Semigroups and Applications to II1

Lecture notes on “Completely positive semigroups and applications ∗ to II1 factors”

Jesse Peterson

May 24, 2013

Contents

1 Completely positive maps 1 1.1 Stinespring’s Dilation Theorem ...... 2 1.2 Bhat’s Dilation Theorem ...... 6 1.3 Poisson boundaries ...... 9

2 The Poisson boundary of a ﬁnite von Neumann algebra 12

3 Derivations 15 3.1 Examples ...... 21 3.1.1 Inner derivations ...... 21 3.1.2 Approximately inner derivations ...... 21 3.1.3 The diﬀerence quotient ...... 22 3.1.4 Group algebras and group-measure space constructions ...... 22 3.1.5 Equivalence relations ...... 23 3.1.6 Free products ...... 24 3.1.7 Free Brownian motion ...... 24 3.1.8 Tensor products ...... 24 3.1.9 q-Gaussians ...... 25 3.1.10 Conditionally negative deﬁnite kernels ...... 26

4 Approximate bimodularity 26

1 Completely positive maps

An operator system E is a closed self adjoint subspace of a unital C∗-algebra A such that ∗ 1 ∈ E. We denote by Mn(E) the space of n × n matrices over E. If A is a C -algebra, then ∗From the minicourse given at the workshop on Operator Algebras and Harmonic Analysis, at Instituto de Ciencias Mathem´aticas

1 ∼ ∗ Mn(A) = A ⊗ Mn(C) has a unique norm for which it is again a C -algebra, where the adjoint given ∗ ∗ ∗ by [ai,j] = [aj,i]. This can be seen easily for C -subalgebras of B(H), and the general case then follows since every C∗-algebra is isomorphic to a C∗-subalgebra of B(H) by the GNS-construction. In particular, if E ⊂ A is an operator system then Mn(E) is again an operator system when viewed ∗ as a subspace of the C -algebra Mn(A). (n) If φ : E → F is a linear map between operator systems, then we denote by φ : Mn(E) → (n) Mn(F ) the map deﬁned by φ ([ai,j]) = [φ(ai,j)]. We say that φ is positive if φ(a) ≥ 0, whenever (n) a ≥ 0. If φ is positive then we say that φ is n-positive and if φ is n-positive for every n ∈ N then we say that φ is completely positive. We’ll say that φ is unital if φ(1) = 1. Throughout these notes we will use the abbreviation u.c.p. for unital completely positive. If φ : E → F is positive then it follows easily that φ is Hermitian, i.e., φ(x∗) = φ(x)∗, for all x ∈ E. Also note that positive maps are continuous. Indeed, if φ : E → F is positive and {xn}n is a sequence which converges to 0 in E, such that limn→∞ φ(xn) = y, then since ω ◦ φ is positive (and hence continuous by the Cauchy-Schwarz inequality) for any state ω ∈ S(B), we then have ω(y) = 0. Since ω was arbitrary it then follows that y = 0 and hence φ is bounded by the closed graph theorem. We also remark that it follows form the Hahn-Banach theorem that any positive linear functional on E extends to a positive linear functional on A which has the same norm. Lemma 1.1. If A and B are unital C∗-algebras and φ : A → B is a unital contraction then φ is positive. Proof. We ﬁrst show that φ is Hermitian. Suppose x = x∗ ∈ A such that φ(x) = a + ib where a, b ∈ B are self-adjoint. Assume kxk ≤ 1. If λ ∈ σ(b) then for all t ∈ R we have (λ + t)2 ≤ kb + tk2 ≤ kφ(x + it)k2 ≤ kx + itk2 ≤ 1 + t2. Hence λ2 + tλ ≤ 1, and as this is true for all t we must then have λ = 0, and hence b = 0. Thus, ω ◦ φ is a state, for any state ω ∈ S(B). Hence if x ≥ 0 then for any state ω we have ω(a) = ω ◦ φ(x) ≥ 0. We must then have a ≥ 0, and hence φ is positive.

1.1 Stinespring’s Dilation Theorem If π : A → B(K) is a representation of a C∗-algebra A and V ∈ B(H, K), then the map φ : A → B(H) given by φ(x) = V ∗π(x)V is completely positive. Indeed, if we consider the operator (n) ⊕n ⊕n (n) V ∈ B(H , K ) given by V ((ξi)i) = (V ξi)i then for all x ∈ Mn(A) we have ∗ φ(n)(x∗x) = V (n) π(n)(x∗x)V (n) = (π(n)(x)V (n))∗(π(n)(x)V (n)) ≥ 0. Generalizing the GNS construction, Stinespring showed that every completely positive map from A to B(H) arises in this way. Theorem 1.2 (Stinespring [Sti55]). Let A be a unital C∗-algebra, and suppose φ : A → B(H), then φ is completely positive if and only if there exists a representation π : A → B(K) and a bounded operator V ∈ B(H, K) such that φ(x) = V ∗π(x)V . We also have kφk = kV k2, and if φ is unital then V is an isometry. Moreover, if A is a von Neumann algebra and φ is a normal completely positive map, then π is a normal representation.

2 ∗ Proof. Consider the sesquilinear form on A ⊗ H given by ha ⊗ ξ, b ⊗ ηiφ = hφ(b a)ξ, ηi, for a, b ∈ A, ⊕n ⊕n ξ, η ∈ H. If (ai)i ∈ A , and (ξi)i ∈ H , then we have

X X X ∗ h ai ⊗ ξi, aj ⊗ ξjiφ = hφ(aj ai)ξi, ξji i j i,j ∗ = hφ((ai)i (ai)i)(ξi)i, (ξi)ii ≥ 0.

Thus, this form is non-negative definite and we can consider Nφ the kernel of this form so that h·, ·iφ is positive definite on K0 = (A ⊗ H)/Nφ. Hence, we can take the Hilbert space completion K = K0. As in the case of the GNS construction, we define a representation π : A → B(K) by first setting ∗ ∗ π0(x)(a ⊗ ξ) = (xa) ⊗ ξ for a ⊗ ξ ∈ A ⊗ H. Note that since φ is positive we have φ(a x xa) ≤ 2 ∗ (n) P 2 2 P 2 kxk φ(a a), applying this to φ we see that kπ0(x) i ai ⊗ ξikφ ≤ kxk k i ai ⊗ ξikφ. Thus, π0(x) descends to a well defined bounded map on K0 and then extends to a bounded operator π(x) ∈ B(K). If we define V0 : H → K0 by V0(ξ) = 1 ⊗ ξ, then we see that V0 is bounded by kφ(1)k and hence extends to a bounded operator V ∈ B(H, K). For any x ∈ A, ξ, η ∈ H we then check that

∗ hV π(x)V ξ, ηi = hπ(x)(1 ⊗ ξ), 1 ⊗ ηiφ

= hx ⊗ ξ, 1 ⊗ ηiφ = hφ(x)ξ, ηi.

Thus, φ(x) = V ∗π(x)V as claimed.

Corollary 1.3 (Kadison [Kad52]). If A and B are unital C∗-algebras, and φ : A → B is u.c.p. then for all x ∈ A we have φ(x)∗φ(x) ≤ φ(x∗x)

Proof. We may assume that B ⊂ B(H). If we consider the Stinespring dilation φ(x) = V ∗π(x)V , then since φ is unital we have that V is an isometry. Hence 1 − VV ∗ ≥ 0 and so we have

φ(x∗x) − φ(x)∗φ(x) = V ∗π(x∗x)V − V ∗π(x)∗VV ∗π(x)V = V ∗π(x∗)(1 − VV ∗)π(x)V ≥ 0.

Lemma 1.4. A matrix a = [ai,j] ∈ Mn(A) is positive if and only if

n X ∗ xi ai,jxj ≥ 0, i,j=1 for all x1, . . . , xn ∈ A.

Pn ∗ Proof. For all x1, . . . , xn ∈ A we have that i,j=1 xi ai,jxj is the conjugation of a by the 1 × n Pn ∗ column matrix with entries x1, . . . , xn, hence if a is positive then so is i,j=1 xi ai,jxj.

3 Pn ∗ Conversely, if i,j=1 xi ai,jxj ≥ 0, for all x1, . . . , xn then for any representation π : A → B(H), and ξ ∈ H we have     * π(x1)ξ π(x1)ξ + n  .   .  X (id ⊗π)(a)  .  ,  .  = hπ(ai,j)π(xj)ξ, π(xi)ξi i,j=1 π(xn)ξ π(xn)ξ *  n  + X ∗ = π  xi ai,jxj ξ, ξ ≥ 0. i,j=1

Thus, if H has a cyclic vector, then (id ⊗π)(a) ≥ 0. But since every representation is decomposed into a direct sum of cyclic representations it then follows that (id ⊗π)(a) ≥ 0 for any representation, and hence a ≥ 0 by considering a faithful representation.

Proposition 1.5. Let E be an operator system, and let B be an abelian C∗-algebras. If φ : E → B is positive, then φ is completely positive.

Proof. Since B is commutative we may assume B = C0(X) for some locally compact Hausdorﬀ space X. If a = [ai,j] ∈ Mn(E) such that a ≥ 0, then for all x1, . . . , xn ∈ B, and ω ∈ X we have     X ∗ X  xi φ(ai,j)xj (ω) =  φ(xi(ω)xj(ω)ai,j) (ω) i,j i,j  ∗   x1(ω) x1(ω)  .   .  = φ  .  a  .  (ω) ≥ 0. xn(ω) xn(ω)

By Lemma 1.4, and since n was arbitrary, we then have that φ is completely positive.

Proposition 1.6. Let A and B be unital C∗-algebras such that A is abelian. If φ : A → B is positive, then φ is completely positive.

Proof. We may identify A with C(K) for some compact Hausdorﬀ space K, hence for n ∈ N we may identify Mn(A) with C(K, Mn(C)) where the norm is given by kfk = supk∈K kf(k)k. Suppose f ∈ C(K, Mn(C)) is positive, with kfk ≤ 1, and let ε > 0 be given. Since K is compact f is uniformly continuous and hence there exists a ﬁnite open cover {U1,U2,...,Um} of K, and a1, a2, . . . , am ∈ Mn(C)+ such that kf(k) − ajk ≤ ε for all k ∈ Uj. Pm For each j ≤ m chose gj ∈ C(K) such that 0 ≤ gj ≤ 1, gj = 1, and gj c = 0. If we j=1 |Uj Pm (n) (n) consider f0 = j=1 gjaj then we have kf − f0k ≤ ε. Therefore we have kφ (f) − φ (f0)k ≤ kφ(n)kε. (n) (n) Since φ (gjaj) = φ(gj)aj ≥ 0, for all 1 ≤ j ≤ m, we have that φ (f0) ≥ 0, and hence since ε > 0 was arbitrary it follows that φ(n)(f) ≥ 0, and it follows that φ is completely positive.

The previous proposition gives us a strengthening of Kadison’s inequality.

4 Corollary 1.7 (Kadison [Kad52]). Let A and B be unital C∗-algebras, and φ : A → B a unital positive map. Then for all x ∈ A normal we have φ(x)∗φ(x) ≤ φ(x∗x).

Proof. Restricting φ to the abelian unital C∗-algebra generated by x we may then assume, by the previous proposition, that φ is completely positive. Hence this follows from Kadison’s inequality for completely positive maps. 0 x∗ Lemma 1.8. Let A be a C∗-algebra, if ∈ (A) is positive, then x = 0, and y ≥ 0. x y M2

Proof. We may assume A is a C∗-subalgebra of B(H), hence if ξ, η ∈ H we have

0 x∗ ξ ξ 2Re(hx∗η, ξi) + hyη, ηi = , ≥ 0. x y η η

The result then follows easily.

Theorem 1.9 (Choi [Cho74]). If φ : A → B is a unital 2-positive map between C∗-algebras, then for a ∈ A we have φ(a∗a) = φ(a∗)φ(a) if and only if φ(xa) = φ(x)φ(a), and φ(a∗x) = φ(a∗)φ(x), for all x ∈ A.

Proof. Applying Kadison’s inequality to φ(2) it follows that for all x ∈ A we have

∗ ∗ ∗ 2! ∗ 2 φ(a a) φ(a x) (2) 0 a (2) 0 a = φ ≥ φ φ(x∗a) φ(aa∗ + x∗x) a x a x φ(a∗)φ(a) φ(a∗)φ(x) = . φ(x∗)φ(a) φ(a)φ(a∗) + φ(x∗)φ(x)

Since φ(a∗a) = φ(a)∗φ(a) it follows from the previous lemma that φ(x∗a) = φ(x∗)φ(a), and φ(a∗x) = φ(a)∗φ(x).

If φ : A → B is u.c.p., then the multiplicative domain of φ is

{a ∈ A | φ(a∗a) = φ(a∗)φ(a) and φ(aa∗) = φ(a)φ(a∗)}.

Note that by Theorem 1.9 the multiplicative domain is a C∗-subalgebra of A, and φ restricted to the multiplicative domain is a homomorphism.

Corollary 1.10. If A is a unital C∗-algebra, φ : A → A is unital and 2-positive, and B ⊂ A is a ∗ C -subalgebra such that φ(b) = b for all b ∈ B then φ is B-bimodular, i.e., for all x ∈ A, b1, b2 ∈ B we have φ(b1xb2) = b1φ(x)b2. Theorem 1.11 (Choi [Cho72]). If A and B are unital C∗-algebras, and φ : A → B is a unital 2-positive isometry onto B, then φ is an isomorphism.

5 Proof. Since a self-adjoint element x of norm at most 1 in a unital C∗-algebra is positive if and only if k1 − xk ≤ 1 it follows that φ−1 is positive. Fix a ∈ A self adjoint, and assume kak ≤ 1. Since φ is onto there exists b ∈ A such that φ(b) = φ(a)2 ≤ φ(a2). Thus b ≤ a2, and since φ−1 is also positive we may apply the previous corollary to the map φ−1 to conclude that a2 = φ−1(φ(a))φ−1(φ(a)) ≤ φ−1(φ(a)2) = b. Hence, φ(a)2 = φ(b) = φ(a2). Since a was an arbitrary self adjoint element, and since A is generated by its self adjoint elements, Theorem 1.9 then shows that φ is an isomorphism.

Exercise 1.12. Show that a C∗-algebra A is abelian if and only if for any C∗-algebra B, every positive map from B to A is completely positive.

1.2 Bhat’s Dilation Theorem Lemma 1.13. If H and K are Hilbert spaces, and V : H → K is a partial isometry, then for A ⊂ B(H), B ⊂ B(K), we have that V ∗ ∗-alg(VBV ∗,A)V = ∗-alg(B,V ∗AV ).

Proof. Using the fact that V ∗V = 1, this follows easily by induction on the length of alternating products for monomials in VBV ∗, and A.

∗ If A0 ⊂ B(H0) is a C -algebra, and φ : A0 → A0 is a unital completely positive map, then one can iterate Stinespring’s dilation as follows:

∗ Lemma 1.14. Suppose A0 ⊂ B(H0) is a unital C -algebra, and φ0 : A0 → A0 is a unital completely positive map. Then there exists a sequence whose entries consist of:

(1) a Hilbert space Hn;

(2) an isometry Vn : Hn−1 → Hn; ∗ (3) a unital C -algebra An ⊂ B(Hn); ∗ (4) a unital representation πn : An−1 → B(Hn), such that πn(An−1), and VnAn−1Vn generate An;

(5) a unital completely positive map φn : An → An; such that the following relationships are satisﬁed for each n ∈ N, x ∈ An−1:

∗ Vn πn(x)Vn = φn−1(x); (1) ∗ Vn AnVn = An−1; (2) φn(πn(x)) = πn(φn−1(x)); (3) ∗ ∗ πn+1(VnxVn ) = Vn+1πn(x)Vn+1. (4)

∗ 00 Moreover, for each n ∈ N we have that the central support of VnVn in An is 1. Also, if A0 is a von Neumann algebra and φ0 is normal then An will also be a von Neumann algebra and πn and φn will be normal for each n ∈ N.

6 Proof. We will first construct the objects and show the relationships (1), (2), and (3) by induction, with the base case being vacuous, and we will then show that (4) also holds for all n ∈ N. So suppose n ∈ N and that (1), (2), and (3) hold for all m < n, (we leave V0 undefined). Recall from the proof of Stinespring’s Dilation Theorem that we may construct a Hilbert space Hn by completing the vector space An−1 ⊗ Hn−1 with respect to the non-negative definite sesquilinear form satisfying ∗ ha ⊗ ξ, b ⊗ ηi = hφn−1(b a)ξ, ηi, for all a, b ∈ An−1, ξ, η ∈ Hn−1. We also obtain a partial isometry Vn : Hn−1 → Hn from the formula

Vn(ξ) = 1 ⊗ ξ, for ξ ∈ Hn−1. We obtain a representation πn : An−1 → B(Hn) (which is normal when A0 is a von Neumann algebra and φ0 is normal) from the formula

πn(x)(a ⊗ ξ) = (xa) ⊗ ξ,

∗ for x, a ∈ An−1, ξ ∈ Hn−1. And recall the fundamental relationship Vn πn(x)Vn = φn−1(x) for all x ∈ An−1, which establishes (1). ∗ ∗ If we let An be the C -algebra generated by πn(An−1) and VnAn−1Vn , then πn : An−1 → An, ∗ ∗ and from Lemma 1.13 we have that Vn AnVn is generated by Vn πn(An−1)Vn and An−1. However, ∗ ∗ Vn πn(An−1)Vn = φn−1(An−1) ⊂ An−1, hence Vn AnVn = An−1, establishing (2). Also, when A0 is a von Neumann algebra and πn is normal it then follows easily that An is then also a von Neumann algebra. ∗ Also note that πn(An−1)VnVn Hn is dense in Hn, and so since πn(An−1) ⊂ An we have that the ∗ 00 central support of VnVn in An is 1. ∗ We then define φn : An → An by φn(x) = πn(Vn xVn), for x ∈ An. This is well defined since ∗ Vn AnVn = An−1, unital, and completely positive. Note that for x ∈ An−1 we have φn(πn(x)) = ∗ πn(Vn πn(x)Vn) = πn(φn−1(x)), establishing (3). Having established (1), (2), and (3) for all n ∈ N, we now show that (4) holds as well. For this, notice first that for a, b ∈ An, x ∈ An−1, and ξ, η ∈ Hn we have

∗ ∗ hπn+1(VnxVn )(a ⊗ ξ), b ⊗ ηi = hVnxVn a ⊗ ξ, b ⊗ ηi ∗ ∗ = hφn(b VnxVn a)ξ, ηi ∗ ∗ ∗ = hπn(Vn b VnxVn aVn)ξ, ηi ∗ = h1 ⊗ πn(xVn aVn)ξ, b ⊗ ηi.

∗ Setting x = 1 and using that Vn+1(1 ⊗ ζ) = ζ for each ζ ∈ Hn, we see that

∗ ∗ ∗ ∗ (Vn+1Vn+1)πn+1(VnVn )(a ⊗ ξ) = (Vn+1Vn+1)(1 ⊗ πn(Vn aVn)ξ) ∗ = 1 ⊗ πn(Vn aVn)ξ ∗ = πn+1(VnVn )(a ⊗ ξ),

7 ∗ ∗ and hence πn+1(VnVn ) ≤ Vn+1Vn+1. If instead we set a = 1 then we have

∗ Vn+1πn(x)ξ = 1 ⊗ πn(x)ξ = πn+1(VnxVn )Vn+1ξ,

∗ ∗ ∗ and so Vn+1πn(x) = πn+1(VnxVn )Vn+1. Multiplying on the right by Vn+1 and using that πn(VnVn ) ≤ ∗ ∗ ∗ Vn+1Vn+1 then gives Vn+1πn(x)Vn+1 = πn+1(VnxVn ). The following theorem was originally proved by Bhat in [Bha99] in the setting of completely positive semigroups, building on work from [Bha96], [BP94], and [BP95]. Other proofs appear in [BS00], [MS02], and Chapter 8 of [Arv03]. We include an elementary proof based on the idea of iterating Stinespring’s dilation [Sti55].

∗ Theorem 1.15 (Bhat [Bha99]). Let A0 ⊂ B(H0) be a unital C -algebra, and φ0 : A0 → A0 a unital completely positive map. Then there exists

(1) a Hilbert space K;

(2) an isometry W : H0 → K; (3) a C∗-algebra B ⊂ B(K); (4) a unital ∗-endomorphism α : B → B;

∗ such that W BW = A0, and for all x ∈ A0 we have

k ∗ k ∗ φ0(x) = W α (W xW )W.

00 Moreover, we have that the central support of P0 in B is 1, and for y ∈ B(K) we have y ∈ B k ∗ k ∗ k ∗ if and only if α (WW )yα (WW ) ∈ α (WA0W ) for all k ≥ 0. Also, if A0 is a von Neumann algebra and φ0 is normal then B will also be a von Neumann algebra, and α will also be normal. Proof. Using the notation from the previous lemma, we may define a Hilbert space K as the directed limit of the Hilbert spaces Hn with respect to the inclusions Vn+1 : Hn → Hn+1. We denote by ∗ Wn : Hn → K the associated sequence of isometries satisfying Wn+1Wn = Vn+1, for n ∈ N, and we ∗ set Pn = WnWn , an increasing sequence of projections. ∗ ∗ ∗ From (2) we have that Pn−1WnAnWn Pn−1 = Wn−1An−1Wn−1, and hence if we define the C - ∗ ∗ algebra B = {x ∈ B(K) | Wn xWn ∈ An, n ≥ 0}, then we have Wn BWn = An, for all n ≥ 0. Also, if A0 is a von Neumann algebra, then so is An for each n ∈ N and from this it follows easily that B is also a von Neumann algebra. We define the unital ∗-endomorphism α : B → B (which is normal when A0 is a von Neumann algebra and φ0 is normal) by the formula

∗ α(x) = lim Wn+1πn+1(W xWn)Wn+1, n→∞ n where the limit is taken in the strong operator topology. Note that α(Pn) = Pn+1 ≥ Pn. From (4) we ∼ see that in general, the strong operator topology limit exists in B, and that for x ∈ An = PnA∞Pn ∗ ∗ the limit stabilizes as α(WnxWn ) = Wn+1πn+1(x)Wn+1.

8 From (1) we see that for n ≥ 0, and x ∈ An we have

∗ ∗ ∗ ∗ Pnα(WnxWn )Pn = WnWn Wn+1πn+1(x)Wn+1WnWn ∗ ∗ = WnVn+1πn+1(x)Vn+1Wn ∗ = Wnφn(x)Wn .

By induction we then see that also for k > 1, and x ∈ A0 we have

k ∗ k−1 ∗ P0α (W0xW0 )P0 = P0α (P0α(W0xW0 )P0)P0 k−1 ∗ = P0α (W0φ0(x)W0 )P0 k ∗ = W0φ0(x)W0 .

00 ∗ By the previous lemma we have that the central support of Pn in WnAnWn is Pn+1. Hence it follows that the central support of P0 in B is 1. Corollary 1.16 (Connes [Con80]). Let M be a countably decomposable properly inﬁnite von Neu- mann algebra and suppose φ : M → M is a normal unital completely positive map, then there exists a (possibly non-unital) ∗-endomorphism α : M → M, and an isometry v ∈ M such that φ(x) = v∗α(x)v, for all x ∈ M. Also, if φ is normal then α is normal as well.

Proof. Bhat’s dilation provides a von Neumann algebra M˜ , a projection p ∈ M˜ with central support 1 such that pMp˜ = M, and a normal (unital) ∗-endomorphism α0 : M˜ → M˜ , such that φ(x) = pα0(x)p, for all x ∈ M. Since M = pMp˜ is property inﬁnite and since p has central support 1 it follows that p and 1 are equivalent in M˜ . Thus there exists a partial isometry v0 ∈ M˜ such ∗ ∗ that v0v0 = 1, and v0v0 = p. If we then consider the possibly non-unital normal ∗-endomorphism α : M → M given by ∗ ∗ ∗ α(x) = v0α0(x)v0 then setting v = v0p we have φ(x) = pα0(x)p = v α(x)v, for all x ∈ M = pMp˜ .

1.3 Poisson boundaries Poisson boundaries of completely positive maps were first defined by Izumi in [Izu02] using the Choi-Effros product from [CE77]. Izumi further developed the theory in [Izu04], and in [Izu12] he credits Arveson with the description of Poisson boundaries as the fixed point algebra of Bhat’s dilation, and this is the perspective we take here. If A ⊂ B(H) is a unital C∗-algebra, and φ : A → A a unital completely positive map, then a projection p ∈ A is said to be coinvariant, if {φn(p)} defines an increasing sequence of projections which strongly converge to 1 in B(H), and such that for y ∈ B(H) we have y ∈ A if and only if φn(p)yφn(p) ∈ A for all n ≥ 0. Note that for n ≥ 0, φn(p) is in the multiplicative domain for φ, and is again coinvariant. We define φp : pAp → pAp to be the map φp(x) = pφ(x)p, then φp is k k normal unital completely positive. Moreover, we have that φp(x) = pφ (x)p for all x ∈ pAp, which can be seen by induction from

k k−1 k k−1 k−1 pφ (x)p = pφ (p)φ (x)φ (p)p = pφ (φp(x))p.

9 Theorem 1.17 (Izumi [Izu12]). Let A ⊂ B(H) be a unital C∗-algebra, φ : A → A a unital completely positive map, and p ∈ A a coinvariant projection. Then the map θ : Har(A, φ) → Har(pAp, φp) given by θ(x) = pxp deﬁnes a completely positive isometric surjection, between Har(A, φ) and Har(pAp, φp). Moreover, if A is a von Neumann algebra and φ is normal then θ is also normal.

Proof. First note that θ is well-deﬁned since if x ∈ Har(A, φ) we have φp(pxp) = pφ(p)xφ(p)p = pxp. Clearly θ is completely positive (and normal in the case when A is a von Neumann algebra and φ is normal). n To see that it is surjective, if x ∈ Har(pAp, φp) then consider the sequence φ (x). For each m, n ≥ 0, we have

m m+n m m n m n m φ (p)φ (x)φ (p) = φ (pφ (x)p) = φ (φp (x)) = φ (x). It follows that {φn(x)} converges in the strong operator topology to an element y ∈ B(H) such that φm(p)yφm(p) = φm(x) for each m ≥ 0, consequently we have y ∈ A. In particular, for m = 0 we have pyp = x. To see that y ∈ Har(A, φ) we use that for all z ∈ A we have the strong operator topology limit lim φ(φn(p)zφn(p)) = φn+1(p)φ(z)φn+1(p) = φ(z), n→∞ and hence φ(y) = lim φ(φm(p)yφm(p)) = lim φm+1(x) = y. m→∞ m→∞ Thus θ is surjective, and since φn(p) converges strongly to 1, and each φn(p) is in the multiplicative domain of φ, it follows that if x ∈ Har(A, φ) then φn(pxp) converges strongly to x and hence kxk = lim kφn(pxp)k ≤ kpxpk ≤ kxk. n→∞ Thus, θ is also isometric.

Corollary 1.18 (Izumi [Izu02]). Let A be a unital C∗-algebra, and φ : A → A a unital completely positive map. Then there exists a C∗-algebra B and a completely positive isometric surjection θ : B → Har(A, φ). Moreover B and θ are unique in the sense that if B˜ is another C∗-algebra, and θ˜ : B˜ → Har(A, φ) is a completely positive isometric surjection, then θ−1 ◦ θ˜ is an isomorphism. Also, if A is a von Neumann algebra and φ is normal, then B is also a von Neumann algebra and θ is normal. Proof. Note that we may assume A ⊂ B(H). Existence then follows by applying the previous theorem to Bhat’s dilation. Uniqueness follows from Theorem 1.11.

We refer to the C∗-algebra B from the previous corollary as the Poisson boundary of φ, and we refer to the map θ as the Poisson transform. Corollary 1.19 (Choi-Eﬀros [CE77]). Let A be a unital C∗-algebra and F ⊂ A an operator system. ∗ If E : A → F is a completely positive map such that E|F = id, then F has a unique C -algebraic structure which is given by x · y = E(xy). Moreover, if A is a von Neumann algebra and F is weakly closed then this gives a von Neumann algebraic structure on F .

10 Proof. When A is a C∗-algebra this follows from Corollary 1.18 since Har(A, E) = F . Also note that since En = E it follows from the proof of Theorem 1.17 that the product structure coming from the Poisson boundary is given by x · y = E(xy). If A is a von Neumann algebra and F is weakly closed then F has a predual F⊥ = {ϕ ∈ A∗ | ϕ(x) = 0, for all x ∈ F } and hence A is isomorphic to a von Neumann algebraic by Sakai’s theorem.

Note that if A is a C∗-algebra, F ⊂ A an operator system, and E : A → F completely positive with E|F = id, then we still have a form of bimodularity for E when we endow F with the Choi- Eﬀros product from Corollary 1.19. In this case though the bimodularity is with respect to two diﬀerent product structures, i.e., we have

E(xay) = x · E(a) · y, for all x, y ∈ F , a ∈ A.

Proposition 1.20. Let A be an abelian C∗-algebra and φ : A → A a normal unital completely positive map. Then the Poisson boundary of φ is also abelian.

Proof. Let B be the Poisson boundary of φ, and let θ : B → Har(A, φ) be the Poisson transform. If C is a C∗-algebra and ψ : C → B is a positive map then θ ◦ ψ : C → Har(A, φ) ⊂ A is positive, and since A is abelian it is then completely positive by Proposition 1.5. Hence, ψ is also completely positive. Since every positive map from a C∗-algebra to B is completely positive it then follows that B is abelian.

Example 1.21. Let Γ be a discrete group and µ ∈ Prob(Γ) a probability measure on Γ such that the support of µ generates Γ. Then on `∞Γ we may consider the normal unital (completely) R −1 positive map φµ given by φµ(f) = µ∗f, where µ∗f is the convolution (µ∗f)(x) = f(g x) dµ(g). ∞ Then Har(µ) = Har(` Γ, φµ) has a unique von Neumann algebraic structure which is abelian by the previous proposition. Notice that Γ acts on Har(µ) by right translation, and since this action preserves positivity it follows from Theorem 1.11 that Γ preserves the multiplication structure as well. Since the support of µ generates Γ, for a non-negative function f ∈ Har(µ)+, we have f(e) = 0 if and only if f = 0. Thus we obtain a natural normal faithful state ϕ on Har(µ) which is given by ϕ(f) = f(e). ∞ ∞ Since ϕ is Γ-equivariant, this extends to a normal u.c.p. mapϕ ˜ : ` Γ o Γ → ` Γ o Γ such that 1 ˜ ϕ˜LΓ = id. It is an easy exercise to see that the Poisson boundary of φ is nothing but the crossed product Har(µ) o Γ. Example 1.22. Let X be a set. A random walk on X is given by transition probabilities µ : X → Prob(X). If f ∈ `∞X we deﬁne the convolution µ ∗ f ∈ `∞X by Z µ ∗ f(x) = f(y) dµ(x, y).

1 ∞ 2 Note that ` Γ o Γ =∼ B(` Γ).

11 Clearly, convolution gives a normal u.c.p. map on `∞X, and hence we may consider the Poisson boundary of this map, which coincides with the boundary of the random walk. If Γ is a discrete group and µ ∈ Prob(Γ) then we recover the previous example by consider the transition probabilities µ˜(gx, x) = µ(g). A particular useful example to consider is when X is a connected locally ﬁnite graph and the transition probabilities for the random walk are given by µ(y, x) = 0, if there is no edge from x 1 to y, and µ(y, x) = d(x) otherwise, where d(x) denotes the degree of x, i.e., the number of edges starting from x.

2 2 Example 1.23. Consider the one sided shift operator s : ` N → ` N given by s(δn) = δn+1. We 2 2 ∗ may then consider the normal u.c.p. map φ : B(` N) → B(` N) given by φ(T ) = sT s . Thinking of 2 operators B(` N) as matrices, we then have that Har(φ) consist of all Toeplitz matrices, i.e., those matrices whose entries hT δn, δmi only depend on n − m. 2 2 If we also consider the two sided shift operators ˜ : ` Z → ` Z, thens ˜ is a unitary and so induces 2 2 ∗ an automorphism α : B(` N) → B(` N) by α(T ) =sT ˜ s˜ . Moreover, if we consider the usual 2 2 2 2 2 embedding ` N ⊂ ` Z, and denote by p the projection onto ` N, then for T ∈ B(` N) = pB(` Z)p we have pα(T )p = φ(T ). Thus, in this case we have an explicit description of Bhat’s dilation. In particular, we have an identiﬁcation of the Poisson boundary of φ with Har(α) = {s˜}0 = 2 LZ ⊂ B(` Z).

2 The Poisson boundary of a ﬁnite von Neumann algebra

Theorem 2.1 (Connes [Con80]). Let N be a ﬁnite von Neumann algebra with a normal faithful trace τ. If φ : N → N is a normal u.c.p. map which preserves τ, then there is a normal Hilbert N-bimodule H, together with a unit vector ξ0 ∈ H such that τ(φ(x)y) = hxξ0y, ξ0i for all x, y ∈ N. Proof. This is very similar to Stinespring’s theorem and we only sketch the proof. First, consider ∗ ∗ on N ⊗alg N the sesquilinear form given by hx ⊗ a, y ⊗ bi = τ(b φ(y x)a). Since φ is completely positive it follows easily that this is non-negative deﬁnite, hence after separation and completion we obtain a Hilbert space H. Since φ is unital and preserves the trace it is then easy to see that the representations x0·(x⊗a) = x0x ⊗ a, and (x ⊗ a) · x0 = x ⊗ ax0, extend to commuting normal (anti-)representations of N on H, so that H is an N-bimodule. Finally, if we set ξ0 = 1 ⊗ 1, then for all x, y ∈ N we have

hxξ0y, ξ0i = hx ⊗ y, 1 ⊗ 1i = τ(φ(x)y).

Corollary 2.2. Let N be a ﬁnite von Neumann algebra with a normal faithful trace τ. If φ : N → N is a normal u.c.p. map which preserves τ, then Har(φ, N) is a von Neumann subalgebra of N.

Proof. If we let H be the normal Hilbert N-bimodule from the previous theorem, and if we take

12 ξ0 ∈ H such that τ(φ(x)y) = hxξ0y, ξ0i, for all x, y ∈ N. Then for x ∈ N we have

2 2 2 ∗ kx − φ(x)k2 = kxk2 + kφ(x)k2 + 2Re(τ(φ(x)x )) 2 ∗ ≤ 2kxk2 + 2Re(τ(φ(x)x )) 2 2 ∗ = kxξ0k + kξ0xk + 2Re(hxξ0x , ξ0i) 2 = kxξ0 − ξ0xk . (5)

2 ∗ And similarly, we have kxξ0 − ξ0xk = 2Re(τ((x − φ(x))x )) ≤ kx − φ(x)k2kxk. Thus we see that Har(φ, N) coincides with {x ∈ N | xξ0 − ξ0x}, and since the latter is clearly a von Neumann subalgebra this then ﬁnishes the proof.

To find interesting examples of boundaries coming from a finite von Neumann algebra N, we see that we should not be looking at u.c.p. maps on N. However, by looking at u.c.p. maps on B(L2(N, τ)) instead we can indeed find interesting examples. Let M be a von Neumann algebra, and let N ⊂ M be a finite von Neumann subalgebra with a normal faithful trace τ. Given a state ϕ ∈ M ∗ we will say that ϕ is a hyperstate if it extends τ. To such a hyperstate we obtain a natural inclusion L2(N, τ) ⊂ L2(M, ϕ) induced from the map 2 2 x1τ 7→ x1ϕ for x ∈ N. Let eN ∈ B(L (M, ϕ)) denote the orthogonal projection onto L (N, τ). We 2 may then consider the u.c.p. map ψϕ : M → B(L (N, τ)), defined by ψϕ(x) = eN xeN . Note that if x ∈ N ⊂ M then we have ψϕ(x) = x. We shall refer to the map ψϕ as the Poisson transform (with respect to ϕ) of the inclusion N ⊂ M. Theorem 2.3. Let M be a von Neumann algebra, and let N ⊂ M be a finite von Neumann subalgebra with a normal faithful trace τ. The correspondence ϕ 7→ ψϕ defined above gives a bijective correspondence between hyperstates on M, and u.c.p., N-bimodular maps from M to B(L2(N, τ)). Moreover, ψϕ is normal if and only if ϕ is normal. Also, this corresondence is a homeomorphism when we consider the space of hyperstates with the weak∗-topology, and the space of u.c.p., N-bimodular maps with the topology of pointwise weak operator topology convergence. Proof. First note that if ϕ is a hyperstate on M, then for all x ∈ M we have

ϕ(x) = hx, 1iτ = hψϕ(x)1, 1iτ .

From this it follows that the correspondence ϕ 7→ ψϕ is one-to-one. To see that it is onto, suppose that ψ : M → B(L2(N, τ)) is u.c.p. and N-bimodular. We deﬁne a state ϕ on M by ϕ(x) = τ ◦ψ(x). For all y ∈ N we then have ϕ(y) = τ ◦ ψ(y) = τ(y), hence ϕ is a hyperstate. Moreover, if y, z ∈ N, and x ∈ M then we have

∗ hψϕ(x)y, ziτ = τ ◦ ψϕ(z xy) (6) ∗ = ϕ(z xy) = hψ(x)y, ziτ , hence, ψϕ = ψ. It is also easy to check that ψϕ is normal if and only if ϕ is. To see that this correspondence is a homeomorphism when given the topologies above, suppose that ϕ is a hyperstate, and ϕα is a net of hyperstates. If we set y = z = 1 in Equation 6, then it

13 follows easily that if ψϕα converges in the pointwise weak operator topology to ψϕ then we have ∗ ∗ that ϕα converges weak to ϕ. Conversely, if ϕα converges weak to ϕ then again using Equation 6 we see that for x ∈ M, and y, z ∈ N we have hψϕα (x)y, ziτ → hψϕ(x)y, ziτ . Since N is dense in 2 L (N, τ), and since kψϕα (x) − ψϕ(x)k ≤ 2kxk, for all α it then follows that ψϕα converges to ψϕ in the pointwise weak operator topology.

When M = B(L2(N, τ)) in the previous theorem then to each hyperstate on B(L2(N, τ)) we obtain a u.c.p. N-bimodular map on B(L2(N, τ)). In particular, composing such maps gives a convolution operation on the space of hyperstates. More generally, if M is a von Neumann algebra, and N ⊂ M is a ﬁnite von Neumann subalgebra with a normal faithful trace τ, then for hyperstates ∗ 2 ∗ ϕ1 ∈ M , and ϕ2 ∈ B(L (N, τ)) we deﬁne the convolution ϕ1 ∗ ϕ2 to be the unique hyperstate on M such that ψϕ1∗ϕ2 = ψϕ1 ◦ ψϕ2 . Lemma 2.4. Using the same notation as above, if Hyp(N ⊂ M) denotes the space of hyperstates on M, then the mapping

Hyp(N ⊂ M) 3 ϕ0 7→ ϕ ∗ ϕ0 ∈ Hyp(N ⊂ M) is continuous in the weak∗-topology. 2 Moreover, if ϕ0 ∈ B(L (N, τ))∗ is a normal hyperstate, then the mapping

2 Hyp(N ⊂ B(L (N, τ))) 3 ϕ0 7→ ϕ0 ∗ ϕ ∈ Hyp(N ⊂ M) is also continuous.

Proof. Since ϕ 7→ ψϕ is a homeomorphism this lemma then follows easily. If ϕ ∈ B(L2(N, τ))∗ then we deﬁne the Poisson boundary B of N with respect to ϕ to be the Poisson boundary of the u.c.p. map ψϕ. From Example 1.21 we see that if Γ is a countable group ∞ and Γy(Y, η) is the action of Γ on a Poisson boundary, then L (Y, η) o Γ is a Poisson boundary for LΓ. Note that we always have a natural inclusion N ⊂ B, since ψϕ is N-bimodular. 2 Restricting the hyperstate ϕ to Har(B(L (N, τ)), ψϕ) we obtain a natural state on B (which we again denote by ϕ), and note that with respect to this state the Poisson transform of the inclusion 2 N ⊂ B agrees with the Poisson transform B → Har(B(L (N, τ)), ψϕ) deﬁned above. Thus, our terminology is consistent in this setting.

Theorem 2.5. Let M be a von Neumann algebra, and let N ⊂ M be a ﬁnite von Neumann 2 subalgebra with a normal faithful trace τ. Let ϕ ∈ B(L (N, τ))∗ be a normal hyper state, and let B be the Poisson boundary of N with respect to ϕ. Then there exists a conditional expectation 2 2 E : B(L (N, τ)) → Har(B(L (N, τ)), ψϕ).

∗ 1 PN ∗k ∗k Proof. Let ϕ0 be a weak -cluster point of { N k=1 ϕ } where ϕ denotes the convolution of ϕ with itself k-times. Then E = ψϕ0 is a cluster point of {ψN } in the topology of pointwise weak 1 PN k operator topology convergence, where ψN = N k=1 ψϕ. 2 2 If x ∈ B(L (N, τ)) then we have kψϕ(ψN (x)) − ψN (x)k ≤ N kxk. Hence it follows that E : 2 2 2 B(L (N, τ)) → Har(B(L (N, τ)), ψϕ). Since, E is the identity on Har(B(L (N, τ)) we then have E2 = E.

14 The trivial case is when ϕe(x) = hx1, 1iτ in which case we have that ψϕe = id, and the Poisson 2 boundary is nothing but B(L (N, τ)). Note that ϕe gives an identity with respect to convolution. Also note that if ϕ ∈ B(L2(N, τ))∗ is a hyperstate, then we have a description of the space of harmonic operators as:

2 2 Har(B(L (N, τ)), ψϕ) = {x ∈ B(L (N, τ)) | ϕ(axb) = ϕe(axb) for all a, b ∈ N}.

2 We’ll say that a normal hyperstate ϕ ∈ B(L (N, τ))∗ is admissible if N is the largest ∗- 2 subalgebra of B(L (N, τ)) which is contained in Har(ψϕ). We’ll say that ϕ is symmetric if ϕ(JxJ) = τ(x) for all x ∈ N. Examples of symmetric admissible hyperstates are easy to ﬁnd. For example, if N is generated by a countable set of unitaries S ⊂ U(N) such that 1 ∈ S, and S = S, then for a fully supported probability measure ν ∈ Prob(S) such that ν(u) = ν(u∗) we may consider the normal u.c.p. map ψ on B(L2(N, τ)) given by Z ψ(x) = (JuJ)x(Ju∗J) dν(u).

Since S generate N, a convexity argument then shows that N is the largest ∗-subalgebra of B(L2(N, τ)) which is contained in Har(ψ). Indeed, if x, x∗x ∈ Har(ψ), then for each a ∈ N we have Z ∗ 2 2 ∗ k (JuJ)x(Ju J)a dν(u)k2 = kψ(x)ak2 = hψ(x x)a, ai Z Z ∗ 2 ∗ 2 = kx(Ju J)ak2 dν(u) = k(JuJ)x(Ju J)ak2 dν(u).

Hence we must have (JuJ)x(Ju∗J) = x for each u ∈ S, and so x ∈ (JNJ)0 = N.

Proposition 2.6. Let N be a ﬁnite von Neumann algebra with a normal faithful trace τ. Let 2 ϕ ∈ B(L (N, τ))∗ be a symmetric admissible hyperstate, and let B be the corresponding Poisson boundary. Then N 0 ∩ B = Z(N), and in particular, when N is a factor then so is B.

Proof. Let θ : B → B(L2(N, τ)) denote the Poisson transform. If x ∈ N 0 ∩ B, then θ(x) ∈ 0 2 N ∩ B(L (N, τ)) = JNJ. Since ϕ is symmetric, ψϕ preserves the trace when restricted to JNJ. Thus Har(ψϕ,JNJ) is a von Neumann subalgebra of JNJ by Corollary 2.2. Since ϕ is admissible we then have x ∈ Har(ψϕ,JNJ) = JNJ ∩ N = Z(N).

3 Derivations

|f(s)−f(t)| Recall that a function f : R → R is Lipschitz if kfkLip = sups,t∈R |s−t| < ∞. We denote by Lip0(R) the set of Lipschitz functions f such that f(0) = 0. 2 2 Lemma 3.1. Suppose f ∈ Lip0(R) and ξ, η ∈ L (N, τ)s.a., then f(ξ), f(η) ∈ L (N, τ) and

kf(ξ) − f(η)k2 ≤ kfkLipkξ − ηk2.

15 Proof. First, let ξ = R sdP (s) be the spectral decomposition for ξ. Then we have Z Z 2 2 2 2 2 |f(s)| d(τ ◦ P )(s) ≤ kfkLip s d(τ ◦ P )(s) = kfkLipkξk2, hence f(ξ) = R f(s)dP (s) ∈ L2(N, τ). To show the above inequality we use Connes’ joint distribution trick [Con76]. We have a state φ on C0(R×R) deﬁned by Σigi ⊗hi 7→ τ(gi(ξ)hi(η)) and hence by the Riesz representation theorem R there exists a Radon measure µ on R×R such that φ(g⊗h) = g(s)h(t)dµ(s, t) for all g, h ∈ C0(R). Since Z lim (s2 + t2)dµ = 0, S,T →∞ s>S,t>T R it follows that we also have φ(g ⊗ h) = g(s)h(t)dµ(s, t) for g, h ∈ Lip0(R). Hence

2 2 2 kf(ξ) − f(η)k2 = τ(f (ξ) − 2f(ξ)f(η) + f (η)) Z Z = f 2(s) − 2f(s)f(t) + f 2(t)dµ(s, t) = |f(s) − f(t)|2dµ(s, t) Z 2 2 2 2 ≤ kfkLip |s − t| dµ(s, t) = kfkLipkξ − ηk2.

A Hilbert N-bimodule is a Hilbert space H together with a pair of commuting representations π : N → B(H), πop : N op → B(H). Given a Hilbert N-bimodule we will denote by xξy the vector π(x)πop(yop)ξ whenever x, y ∈ N and ξ ∈ H. We’ll say that the bimodule is normal if the representations π and πop are normal. We’ll say that the bimodule is symmetric if there exists an antilinear involution J on H such that J (xξy) = y∗J (ξ)x∗ for all x, y ∈ N and ξ ∈ H. The trivial bimodule is L2(N, τ) where the bimodule structure is given by left or right multiplication. We will also call L2(N, τ)⊗L2(N, τ) the coarse bimodule, where the bimodule structure is given by π(x)ξ = (x ⊗ 1)ξ, and πop(yop)ξ = ξ(1 ⊗ y). If H is a Hilbert N-bimodule, then a closable derivation is a densely deﬁned closable operator 2 δ : L (N, τ) → H such that D(δ) ⊂ nτ is a ∗-algebra and δ(xy) = xδ(y) + δ(x)y for all x, y ∈ D(δ). We’ll say that a derivation is symmetric if H is symmetric and we have J (δ(x)) = δ(x∗) for all x ∈ D(δ). Recall that if T : K → K is a closable operator and ξk ∈ D(T ) such that ξk → ξ and K = lim supk kT (ξk)k < ∞ then ξ ∈ D(T ), and kT (ξ)k ≤ K. Indeed, if T is closable then Graph(T ) ⊂ K ⊕K is a subspace, and since (ξk,T (ξk)) ∈ Graph(T ) is bounded, it has a weak limit cluster point. Theorem 3.2 (Sauvageot [Sau90], Davies-Lindsay [DL92]). Let H be a Hilbert N-bimodule, let 2 δ : L (N, τ) → H be a closable symmetric derivation. If ξ ∈ D(δ)s.a., and f ∈ Lip0(R), then f(ξ) ∈ D(δ) and kδ(f(ξ))k ≤ kfkLipkδ(ξ)k. Moreover, If δ is closed as an operator from N with the uniform topology to H, x ∈ D(δ)s.a., and f ∈ Lip0(R) then f(x) ∈ D(δ).

16 Proof. First consider the case ξ = x ∈ D(δ) ⊂ N. If we let K = σ(x) then the representations of ∼ ∗ ∗ N in B(H) gives rise to a representationπ ˜ of C(K × K) = C (x) ⊗max C (x) in B(H) such that π˜(Σigi ⊗ hi)ξ = Σigi(x)ξhi(x). Given f ∈ C1(K) denote by f˜ ∈ C(K × K) the function ( f(s)−f(t) if s 6= t; f˜(s, t) = s−t f 0(s) otherwise.

If we are given the polynomial p(t) = tn then we have

n−1 k n−1−k δ(p(x)) = Σk=0x δ(x)x n−1 k n−1−k = Σk=0π(s t ) · δ(x) =π ˜(p) · δ(x). By linearity we therefore have that δ(p(a)) = π(˜p) · δ(a) for all polynomials p with vanishing scalar coeﬃcient. Taking a polynomial approximation of f ∈ C1(K) it follows that f(x) ∈ D(δ) and ˜ ˜ δ(f(x)) = π(f) · δ(x). In particular we have that kδ(f(a))k ≤ kfk∞kδ(a)k = kfkLipkδ(a)k for all 1 f ∈ C (R). If now f ∈ Lip0(R) then by taking a sequence of functions ϕn ≥ 0 with support [−1/n, 1/n] 1 R 1 such that ϕn ∈ C (R), and ϕndµ = 1, it is then easy to show that f ∗ ϕn ∈ C (R), f ∗ ϕn → f uniformly on compact sets, and kf ∗ ϕnkLip ≤ kfkLip. Hence it follows that f(x) ∈ D(δ) and

kδ(f(x))k ≤ lim sup kδ((f ∗ ϕn)(x))k n

= lim sup kπ˜(f^∗ ϕn) · δ(x)k n

≤ lim sup kf ∗ ϕnkLipkδ(x)k ≤ kfkLipkδ(x)k. n

Since f ∗ ϕn → f uniformly on K it follows that k(f ∗ ϕn)(x) − f(x)k∞ → 0 and since each (f ∗ ϕn) can be uniformly approximated in C1(K) by polynomials it follows that if δ is closed as an operator from N with the uniform topology to H then we have f(x) ∈ D(δ). If we consider now ξ ∈ D(δ)s.a. then there exists a sequence xn ∈ D(δ)s.a. such that kξ−xnk2 → 0 and kδ(ξ) − δ(xn)k → 0. By Lemma 3.1 kf(ξ) − f(xn)k2 → 0 and it therefore follows that f(ξ) ∈ D(δ) and kδ(f(ξ))k ≤ lim sup kδ(f(xn))k n

≤ lim sup kfkLipkδ(xn)k = kfkLipkδ(ξ)k. n

Notation 3.3. Let H be a Hilbert N-bimodule, let δ : L2(N, τ) → H be a closable symmetric derivation. We introduce the following operators on L2(N, τ): ∆ := δ∗δ;

φt := exp −t∆; α ρ := ; α α + ∆

17 It follows from general theory that the above maps also give corresponding operators on N for which we will use the same notation. Using the spectral theorem for unbounded operators we have the following relationships between these maps (all limits are in the sense of pointwise convergence):

φt ◦ φs = φt+s;

(β − α)ρα ◦ ρβ = βρα − αρβ; 1 ∆ = lim (id − φt) = lim α(id − ρα); t→0 t α→∞ φt = exp(−t∆) = lim exp(−tα(id − ρα)), α→∞ Z ∞ −1 −αt ρα = α(α + ∆) = α e φtdt. 0 Theorem 3.4 (Sauvageot [Sau90]). Let H be a Hilbert N-bimodule, let δ : L2(N, τ) → H be a closable symmetric derivation. Then the maps φt, and ρα are both contractive, τ-symmetric, and completely positive. ∞ (tα)n n Proof. Since limα→∞ exp(−tα(id − ρα)) = limα→∞ Σn=0 n! ρα it is enough to show that the ρα’s are completely positive. Also, by scaling δ it is enough to show that ρ1 is completely positive. (n) 2 2 (n) Moreover, if we consider δ : L (Mn(C)⊗, tr ⊗ τ) → L (Mn(C), tr)⊗H such that D(δ ) = (n) (n) 1 n( ) ⊗ D(δ), and δ (A ⊗ x) = A ⊗ δ(x). Then it is not hard to see that ρ = . M C 1 1+δ(n)∗δ(n) Hence, it is enough to show that ρ1 is positive. 2 Suppose η ∈ L (N, τ) such that η ≥ 0, and denote ξ = ρ1(η) ∈ D(∆) ⊂ D(δ) so that ξ +∆(ξ) = η. Since δ is symmetric it is easy to see that ξ is self-adjoint. Denote by I : D(δ) → L2(N, τ) the identity map, and note that if we endow D(δ) with the 2 2 2 graph norm kxkG = kxk2 + kδ(x)k then we have that D(δ) is a Hilbert space, I is a bounded ∗ operator and ρ1 = I . By hypothesis we have that kδ(|ξ|)k ≤ kδ(ξ)k and hence k|ξ|kG ≤ kξkG. Also, note that since η ≥ 0 we have η1/2ξη1/2 ≤ η1/2|ξ|η1/2 and hence |hξ, ηi| ≤ h|ξ|, ηi. Therefore we have 2 ∗ kξkG = |hξ, I ηiG| = |hξ, ηi| ≤ h|ξ|, ηi ∗ 2 = h|ξ|,I ηiG = h|ξ|, ξiG ≤ k|ξ|kGkξkG ≤ kξkG. 2 Thus k|ξ|kG = h|ξ|, ξiG = kξkG and hence ξ = |ξ|. Lemma 3.5. Let H be a Hilbert N-bimodule, let δ : L2(N, τ) → H be a closable symmetric derivation. If a ∈ D(δ), then |a| ∈ D(δ) and kδ(|a|)k2 + kδ(|a∗|)k2 ≤ kδ(a)k2 + kδ(a∗)k2. (2) 2 ∼ ⊕4 (2) Proof. Consider δ : M2(L (N, τ)) → M2(H) = H . Then δ is a closable symmetric derivation (2) 0 a (2) with δ(2) = δ . If a ∈ D(δ) then ∈ D(δ ) and hence by Theorem 3.2 we have that a∗ 0 0 a |a| 0 (2) = ∈ D(δ ), a∗ 0 0 |a∗| and kδ(|a|)k2 + kδ(|a∗|)k2 ≤ kδ(a)k2 + kδ(a∗)k2.

18 Lemma 3.6. Let H be a Hilbert N-bimodule, let δ : L2(N, τ) → H be a closable symmetric derivation. Suppose that δ is closed as an operator from N with the uniform topology to H. If x ∈ D(δ)s.a. ∩N then there exists a sequence xn ∈ D(δ)s.a. such that kxn −xk2 → 0, kδ(xn)−δ(x)k2 → 0, and kxnk∞ ≤ kxk∞ for all n ∈ N.

Proof. Note that from Theorem 3.2 we have that D(δ)s.a. is closed under Lipschitz functional calculus. Let f(t) = t ∧ kxk∞ ∨ (−kxk∞), and suppose yn ∈ D(δ)s.a. such that kyn − xk2, and kδ(yn) − δ(x)k → 0. By Lemma 3.1 we have that kf(yn) − xk2 = kf(yn) − f(x)k2 → 0. Also,

lim sup kδ(f(yn))k = kδ(x)k < ∞. n

Hence a subsequence of f(yn) converges weakly to x in the graph norm on D(δ), and so by taking convex combinations of f(yn) we obtain the desired sequence. Theorem 3.7 (Davies-Lindsay [DL92]). Let H be a Hilbert N-bimodule, and let δ : L2(N, τ) → H be a closable symmetric derivation. Then D(δ) ∩ nτ is a ∗-subalgebra. If H is normal then we have that δ is again a derivation. |D(δ)∩nτ Proof. If we consider ﬁrst the closure of δ as an operator from N with the uniform topology to H, then it follows easily from the triangle inequality that the domain of this closure is again a ∗-algebra and the extension of δ to this domain is again a derivation. Thus we may assume that δ is closed as an operator from N to H. 2 2 If a ∈ D(δ)∩nτ then since t 7→ t ∧kak is Lipschitz, it from Lemma 3.5 together with Lemma 3.1 2 that |a| ∈ D(δ). Now if x, y ∈ D(δ) ∩ nτ then we may apply the polarization identity to conclude

1 √ j √ j x∗y = Σ3 −1 |x + −1 y|2 ∈ D(δ). 4 j=0 The fact that δ is again a derivation if H is a normal bimodule follows by applying |D(δ)∩nτ Lemma 3.6 together with the triangle inequality.

Remark 3.8. In the previous theorem in order to show that D(δ)∩nτ was again a ∗-subalgebra, we (2) used only the facts that the quadradic form q(x) = kδ(x)k2 and its ampliﬁcation q(2)(x) = kδ (x)k2 satisfy the properties that the domain D(δ) is self-adjoint, closed under taking absolute values and Lipschitz functional calculus on self-adjoint elements, and satisﬁes

q(n)(x∗) = q(n)(x);

q(n)(|x|) + q(n)(|x∗|) ≤ q(n)(x) + q(n)(x∗); q(n)(f(y)) ≤ kfk2 q(n)(y), Lip0 ∗ where n = 1, 2, x, y ∈ D(δ) with y = y , and f ∈ Lip0(R).

Theorem 3.9 (Sauvageot [Sau89]). If φt : N → N is a strongly continuous semigroup of τ- symmetric, unital, completely positive maps then there is a Hilbert N-bimodule H and a densely 2 ∗ deﬁned closable symmetric derivation δ : L (N, τ) → H such that φt = exp(−tδ δ), for all t ≥ 0.

19 2 −t∆ Proof. By the Hille-Yoshida theorem there is a positive operator ∆ on L (N, τ) such that φt = e for t > 0, where 2 D(∆) = {ξ ∈ L (N, τ) | lim kξ − φt(ξ)k2 < ∞}, t→0 ∗ and k∆ξk2 = limt→0 kξ − φt(ξ)k2 for ξ ∈ D(∆). We will show that ∆ is of the form δ δ for a closable symmetric derivation δ. By Theorem 2.1 to each φt we can associate a bimodule Ht, and a vector ξt ∈ Ht such that τ(φt(x)y) = hxξty, ξti for each x, y ∈ N. Also, note that since φt is τ-symmetric, there exist ∗ ∗ an anti-linear isometric involution J on Ht given by J(xξty) = y ξtx . Consider the derivation, 2 δt : L (N, τ) → Ht given by D(δt) = nτ , and

δt(x) = xξt − ξtx.

It is then easy to check that δt is a closable symmetric derivation for each t > 0. Note that 1/2 2 1 1/2 2 D(∆ ) = {ξ ∈ L (N, τ) | lim k(id − φt) (ξ)k2 < ∞}, t→0 t and we have 1/2 2 1 1/2 2 1 2 k∆ ξk2 = lim k(id − φt) (ξ)k2 = lim kδt(ξ)k , t→0 t t→0 t for ξ ∈ D(∆1/2). By taking a limit, it then follows from Lemma 3.5, and Theorem 3.2, that for x ∈ D(∆1/2) we have |x| ∈ D(∆1/2) with

1/2 2 1/2 ∗ 2 1/2 2 1/2 ∗ 2 k∆ (|x|)k2 + k∆ (|x |)k2 ≤ k∆ (x)k2 + k∆ (x )k2, ∗ 1/2 1/2 and if x = x ∈ D(∆ ), and f ∈ Lip0(R), then f(x) ∈ D(∆ ) and 1/2 1/2 k∆ (f(x))k2 ≤ kfkLipk∆ (x)k2. By Remark 3.8 we then have that D(∆1/2) is a ∗-subalgebra. 1/2 1/2 On D(∆ ) ⊗alg D(∆ ) we may then deﬁne the sesquilinear form

1 ∗ ∗ ∗ hy ⊗ b, x ⊗ ai = lim τ(a (φt(x y) − φt(x )φt(y))b) t→0 t = −τ(a∗(∆(x∗y) − ∆(x∗)y − x∗∆(y)).

This is well defined since x∗y ∈ D(∆1/2) ⊂ D(∆). Also, this is non-negative definite which can be (n) seen by applying Kadison’s inequality for the u.c.p. maps φt , and then taking a limit. We then obtain a Hilbert space H by separation and completion. Moreover, we have an anti-linear isometric involution J on H given by J(x ⊗ a) = a∗ ⊗ x∗ − a∗x∗ ⊗ 1. We may define a right N-module structure to H by the formula (x ⊗ a) · z = x ⊗ az, and then using J we may define a right N-module structure by the formula z · (x ⊗ a) = zx ⊗ a − z ⊗ xa.A simple check shows that this gives a well defined bimodule structure to H. 1/2 Finally, we define the derivation δ : D(∆ ) ∩ nτ → H by the formula 1 δ(x) = √ (x ⊗ 1). 2

20 It is easy to check that this is indeed a symmetric derivation and we have 1 hδ(x), δ(y)i = hx ⊗ 1, y ⊗ 1i 2 1 = τ(∆(y∗)x + y∗∆(x) − ∆(y∗x)) 2 = h∆1/2(x), ∆1/2(y)i, which shows that δ is closable and ∆ = δ∗δ.

3.1 Examples We now list some examples of closable derivations which appear in von Neumann algebras. Many of these examples overlap. For many of these examples the corresponding semigroups φt can be computed explicitly. It is a good exercise to try to do so when possible. Note that for verifying closability we will use repeatedly the well known fact that an operator D : H → H is closable if and only if the adjoint D∗ is densely deﬁned.

3.1.1 Inner derivations Let H be a normal symmetric Hilbert N-bimodule and let ξ ∈ H be given such that J (ξ) = ξ, 2 then one can construct a inner derivation whose domain is nτ ⊂ L (N, τ) given by δ(x) = xξ − ξx.

A vector η ∈ H is said to be left (resp. right) bounded if there is a constant C > 0 such that kxηk ≤ Ckxk2kηk (resp. kηxk ≤ Ckxk2kηk). Given any η ∈ H if we consider the normal positive linear functional ψ on N given by ψ(x) = hxη, ηi then by duality ψ corresponds to a positive element 0 1 η ∈ L (N, τ). Thus there exists a sequence of projections pn ∈ N which converge σ-weakly to the 0 identity such that pnη ∈ N for all n and hence pnη is left bounded for all n ∈ N. The same argument holds for right boundedness and it therefore follows that the set of vectors in H which are both left and right bounded is dense in H. Since D(δ∗) contains the set of left and right bounded vectors it follows that δ is closable.

3.1.2 Approximately inner derivations Suppose that N is separable and that we have a countable generating set of self-adjoint elements x1, x2,... Also, suppose Hj is a sequence of symmetric normal Hilbert N-bimodules and ξi ∈ Hi are given such that J (ξi) = ξi for all i ∈ N, and kxξi − ξixk → 0 for all x ∈ N. Then by taking a 2 i subsequence we may assume that kxjξi − ξixjk < 1/2 for all 1 ≤ j ≤ i. We may therefore deﬁne a derivation δ on the algebra A generated by x1, x2,... given by

δ(a) = ⊕i∈N(aξi − ξia), for all a ∈ A. 0 If Hi denotes the set of vectors in Hi which are both left and right bounded, then it is easy to ∗ j 0 see that D(δ ) contains the dense subset ∪j∈N(⊕i=1Hi ) ⊂ ⊕i∈N Hi and hence δ is closable.

21 If we consider Pi the projection from Hi to the space of N central vectors in Hi then it is easy 2 to see that δ will be inner if and only if Σi∈Nkξi − Pi(ξi)k < ∞. A ﬁnite von neumann algebra B has property (T) if and only if no such sequence ξi exists with 2 Σi∈Nkξi − Pi(ξi)k = ∞, thus we see that if B is separable II1 factor which does not have property (T) then for any weakly dense, countably generated unital ∗-subalgebra D ⊂ B there exists a non-inner symmetric closable derivation δ : L2(N, τ) → H into some normal Hilbert B-bimodule H such that D ⊂ D(δ). The converse of the previous paragraph also holds but is a bit more subtle to establish.

3.1.3 The diﬀerence quotient

∞ 2 If N = L (R, µ) where µ is a diﬀuse Radon probability measure on R. Then L (R × R, µ × µ) is ∞ a normal Hilbert L (R, µ)-bimodule where the bimodule structure is given by

(f · g · h)(s, t) = f(s)g(s, t)h(t).

2 2 We obtain a derivation δ : L (R, µ) → L (R × R, µ × µ) by f(s) − f(t) δ(f)(s, t) = . s − t Note that we do not need to define δ(f) when s = t since this set has measure 0. The domain of δ ∞ f(s)−f(t) 2 is defined to be {f ∈ L (R, µ) | s−t ∈ L (R × R, µ × µ). 2 F (s,t) 2 ∗ Note that if F ∈ L (R × R, µ × µ) such that s−t ∈ L (R × R, µ × µ) then F ∈ D(δ ). It is easy to see that such functions are dense and hence δ is closable. For a specific example if we consider the case when µ(x) = √1 exp(− 1 x2)dx is the Gaussian 2π 2 2 measure. Then the function f(t) = t is in L (R, µ) and it is easy to see that f ∈ D(δ) and δ(f)(s, t) = 1.

3.1.4 Group algebras and group-measure space constructions Let Γ be a group and let π :Γ → O(K) be an orthogonal representation. We have a pair of 2 commuting representations of Γ on H = K⊗R` Γ given by π ⊗ λ and 1 ⊗ ρ. The representation 1 ⊗ ρ can also be viewed as a representation of Γop via the isomorphism gop 7→ g−1. By Fell’s absorption lemma both of these representations are equivalent to a multiple of the left regular representation and hence extend to normal representations of the group von Neumann algebras LΓ and LΓop. Thus H is a normal Hilbert LΓ-bimodule, which is symmetric via the involution

J (ξ ⊗R Σg∈Γαgδg) = ξ ⊗R Σg∈Γαgδg−1 . A 1-cocycle is a map c :Γ → K such that c(gh) = c(g) + π(g)c(h) for all g, h ∈ Γ. Associated 2 to this 1-cocycle is a derivation δ : L (LΓ, τ) → H which is deﬁned on the group algebra CΓ ⊂ LΓ by δ(Σg∈Γαgug) = Σg∈Γαgc(g) ⊗ δg. The derivation property as well as the symmetry is readily veriﬁed and it is easy to see that the adjoint of δ contains the dense subspace K ⊗alg CΓ, hence δ is closable. More generally, if we also have a measure preserving action Γy(X, µ), and we denote by σ : Γ → U(L2(X, µ)) the Koopman representation, then we again have commuting representations on

22 L2(X, µ)⊗K⊗`2Γ given by σ ⊗ π ⊗ λ and 1 ⊗ 1 ⊗ ρ. Moreover, these representations extend to turn 2 2 ∞ L (X, µ)⊗K⊗` Γ into a normal Hilbert L (X, µ) o Γ-bimodule. 2 ∞ 2 2 We can then deﬁne a derivation δ : L (L (X, µ) o Γ) → L (X, µ)⊗K⊗` Γ which is deﬁned by letting δ|L∞(X,µ) = 0, and δ(ug) = 1 ⊗ c(g) ⊗ δg, and then extending to the algebra generated by ∞ L (X, µ) and CΓ.

3.1.5 Equivalence relations Let R be a countable discrete measure preserving equivalence relation. Letµ ˜ be the infinite measure R on R given byµ ˜(E) = X |{(x, y) ∈ E}|dµ(x). 2 Given f, g ∈ L (R, µ˜) we define the convolution of f and g as (fg)(x, y) = Σz∼xf(x, z)g(z, y). In general, the convolution of f and g will no longer live in L2(R, µ˜), although by the Cauchy-Schwarz inequality it is well defined as a measurable function on R. If f is the characteristic function on the diagonal of X×X then fg = gf = g for all g ∈ L2(R, µ˜). Also, we have an involution on L2(R, µ˜) given by f ∗(x, y) = f(y, x). If fg ∈ L2(R, µ˜) for all g ∈ L2(R, µ˜) then it can be shown using the closed graph theorem that f is a bounded operator. We may thus consider

LR = {f ∈ L2(R, µ˜) | D(f) = L2(R, µ˜)} ⊂ B(L2(R, µ˜)).

This turns out to be a von Neumann algebra, and is in fact ﬁnite since we have a trace given by Z τ(f) = f(x, x)dµ(x).

L∞(X, µ) is naturally a von Neumann subalgebra of LR by considering the functions whose support is on the diagonal of X × X. An orthogonal representation of R consists of a Borel ﬁeld of real Hilbert spaces K = (xK, x ∈ X) together with a Borel map π : R → O(K) such that π(x, y) ∈ O(yK, xK) and π(x, y)π(y, z) = π(x, z) for all x ∼ y ∼ z. 2 Given a representation K = (xK, x ∈ X) of R consider the Hilbert space H = L (R, K, µ˜) of µ˜-square integrable functions from R to K ⊗R C such that ξ(x, y) ∈ xK ⊗R C. We can then consider a normal action of LR on H given by

(fξ)(x, y) = Σz∼xf(x, z)π(x, z)ξ(z, y).

We also have a commuting normal right action of LR on H given by

(ξf)(x, y) = Σz∼xξ(x, z)f(z, y) = Σz∼xf(z, y)ξ(x, z).

A cocycle of R is a Borel map c : R → K such that c(x, y) ∈ xK for all x ∼ y and

c(x, z) = c(x, y) + π(x, y)c(y, z), for all x ∼ y ∼ z. Given a cocycle we may consider the Borel map δ(f): R → H by

δ(f)(x, y) = f(x, y)c(x, y)

23 This map will not in general be in H since it need not be square integrable, however this does map into H for a weakly dense subalgebra D(δ) of LR. If f, g ∈ D(δ) then we have

δ(fg)(x, y) = Σz∼xf(x, z)c(x, y)g(z, y)

= Σz∼xf(x, z)c(x, y)g(z, y) = Σz∼xf(x, z)(c(x, z) + π(x, z)c(z, y))g(z, y) = δ(f)g + fδ(g). Hence, δ gives a derivation. It is also not too diﬃcult to see that δ is both symmetric and closable. Note also that δ|L∞(X,µ) = 0.

3.1.6 Free products

If Mj are ﬁnite von Neumann algebras for j = 1, 2 and B ⊂ Mj is a von Neumann subalgebra such that the traces on M1 and M2 agree on B, then B ⊂ M1 ∗B M2 and we have a derivation 2 2 δ : L (M1 ∗B M2) → L hM1 ∗B M2, eBi which is given by

xe − e x if x ∈ M ; δ(x) = B B 1 0 if x ∈ M2.

Since beB = eBb for all b ∈ B, we can use freeness to extend δ to the algebraic free product, and it’s not hard to see that this gives a closable symmetric derivation.

3.1.7 Free Brownian motion

Let M be a finite von Neumann algebra generated by self-adjoint elements x1, . . . , xm. Let s1, . . . , sm be freely independent semi-circular operators which are also freely independent of x1, . . . , xm. Let t > 0 and define Nt to be the von Neumann algebra generated by x1 + ts1, . . . , xm + tsm. Such operators are algebraically free and hence for each 1 ≤ k ≤ m we can define a unique derivation 2 2 δk : L (Nt, τ) → L (Nt⊗Nt, τ ⊗ τ) whose domain is ∗ − alg(x1 + ts1, . . . , xm + tsm) and such that δk(xj + tsj) = δj,k1 ⊗ 1. ∗ With this definition Voiculescu showed that we always have D(δk) ⊗alg D(δk) ⊂ D(δk) and thus δk is a closable derivation.

3.1.8 Tensor products

2 Let Ni, be a finite von Neumann algebra for i ∈ N and suppose that δi : L (Ni, τ) → Hi is a closable symmetric derivation in to a normal Hilbert Ni-bimodule for each i ∈ N. Let N = ⊗i∈NNi, and for each k ∈ N define the normal Hilbert N-bimodule ˜ 2 Hk = (⊗i∈N,i6=kL (Ni, τ))⊗Hi. ˜ 2 We can then define a derivation δk : L (N, τ) → H˜k whose domain is the algebraic tensor product ⊗i∈ND(δi) by ˜ δk(⊗i∈Nxi) = (⊗i∈N,i6=kxi) ⊗ δk(xk).

24 ˜ 2 Taking the direct sum we can consider H = ⊕k∈NHk and the derivation δ : L (N, τ) → H given by ˜ δ(x) = ⊕k∈Nδk(x). i Then δ is a symmetric closable derivation. Moreover, if φt is the semigroup of completely positive maps associated to δi then we can see that the semigroup of completely positive maps associated i to δ is ⊗i∈Nφt.

3.1.9 q-Gaussians

Let K0 be a real Hilbert space and let K be its complexiﬁcation. If q ∈ [−1, 1] is given, then we can deﬁne a sesquilinear form h·, ·iq on the sum of the algebraic tensor products

⊗2 CΩ ⊕ K ⊕ K ⊕ · · · , such that ι(π) n hξ1 ⊗ · · · ⊗ ξm, η1 ⊗ · · · ⊗ ηniq = δm,nΣπ∈Sn q Πk=1hξk, ηπ(k)i, where ι(π) = |{(i, j) | 1 ≤ i < j ≤ n, π(i) > π(j)}|. This is a non-negative definite form (it is even positive definite if q 6= −1, 1) and the corresponding Hilbert space Fq(K) is called the q-Fock space. ∗ For each ξ ∈ K0 we obtain the creation and annihilation operators a(ξ) , a(ξ): Fq(K) → Fq(K) given by a(ξ)∗(Ω) = ξ; ∗ a(ξ) (η1 ⊗ · · · ⊗ ηn) = ξ ⊗ η1 ⊗ · · · ⊗ ηn; a(ξ)(Ω) = 0; n i−1 a(ξ)(η1 ⊗ · · · ⊗ ηn) = Σi=1q hξ, ηiiη1 ⊗ · · · ⊗ ηî ⊗ · · · ηn, whereη î denotes omission of ηi. The operators a(ξ) and a(ξ)∗ are bounded when q < 1, and we define the q-Gaussian operator x(ξ) = a(ξ) + a(ξ)∗. Note that when q = 1 we can still interpret the above operators as unbounded operators and in this case all the operators commute and x(ξ) has Gaussian distribution with mean 0 and variance kξk2. The set of q-Gaussian operators generate a finite von Neumann algebra Γq(K0) which has a 2 trace given by τ(x) = hxΩ, Ωi, and in this way we can identify L (Γq(K0)) with Fq(K). 2 By embedding K0 as K0 ⊕ 0 ⊂ K0 ⊕ K0, we have that L (Γq(K0 ⊕ K0)) is a normal Hilbert Γq(K0)-bimodule. We obtain a derivation

2 2 δ : L (Γq(K0)) → L (Γq(K0 ⊕ K0)) by deﬁning δ(x(ξ)) = 0 ⊕ x(ξ) for al ξ ∈ K0, and then using the derivation property to extend to 2 the ∗-algebra generated by the x(ξ)’s. If we view L (Γq(K0)) as Fq(K) via the embedding x 7→ xΩ, then we can compute δ directly as

n δ(ξ1 ⊗ · · · ⊗ ξn) = Σk=1(ξ1 ⊕ 0) ⊗ · · · (ξk−1 ⊕ 0) ⊗ (0 ⊕ ξk) ⊗ (ξk+1 ⊕ 0) ⊗ · · · ⊗ (ξn ⊕ 0).

25 2 Actually, the Γq(K0)-bimodule that δ maps into is the sub-bimodule H of L (Γq(K0 ⊕ K0)) which is generated by vectors of the form (0 ⊕ ξ) for ξ ∈ K. Depending on q and dim(K0) the bimodule H will have diﬀerent properties. For example, when q = 0 then it is not hard to check that H, as a bimodule is isomorphic to a direct sum of coarse bimodules.

3.1.10 Conditionally negative definite kernels Let S be a set, let K be a real Hilbert space and let q : S → K. The function on S × S given by (s, t) 7→ kq(s) − q(t)k2 is called a conditionally negative definite kernel. We denote by N the semifinite von Neumann algebra B(`2S). We also identify N with B(`2S)⊗ 2 2 2 1 ⊂ B(` S⊗RK), in this way the Hilbert-Schmidt operators HS(` S, ` S⊗K) gives a Hilbert N- bimodule, where the bimodule structure is given by composition of operators. 2 2 Associated with q is the operator Mq : ` S → ` S⊗K whose domain is the algebraic sum CS, and on this it is given by Mq(δs) = δs ⊗ q(s). Note that this operator is closable since it is easy to ∗ ∗ see that CS ⊗alg K ⊂ D(Mq ) and hence Mq is densely defined. It is also self-adjoint since K is a real Hilbert space. We also remark that if one can think about operators from `2S to `2S⊗K as “K valued S × S matrices”, and from this perspective we see that Mq is just the diagonal matrix with diagonal entries q(s) (this perspective also gives an anti-linear isometry J by taking the adjoint of a matrix). We can then define a derivation δ : L2(N, τ) = HS(`2S) → HS(`2S, `2S⊗K) whose domain is the finite rank operators, and such that

δ(A) = AMq − MqA.

Note that if A ∈ FR(`2S), and B ∈ FR(`2S, `2S⊗K) then we have

∗ |hδ(A),Bi| = |τB(`2S⊗K)((AMq − MqA)B )|

∗ ∗ ∗ ∗ = |τB(`2S⊗K)(AMqB ) + τB(`2S)(B MqA)| ≤ kAkHS(kMqB kHS + kB MqkHS). This shows that FR(`2S, `2S⊗K) ⊂ D(δ∗) and so in particular δ is closable.

4 Approximate bimodularity

For the sequel N will be a ﬁnite von Neumann algebra with trace τ. We will also assume that 1 ∈ D(δ), and hence φt and ρα will be unital completely positive maps. Notation 4.1. We will need still more notation. Given a closable symmetric derivation δ : L2(N, τ) → H we continue to denote by ρ = α , we will additionally deﬁne α α+δ∗δ √ ζα := ρα; ∆ θ := id −α−1/2∆1/2 ◦ ζ = id −( )1/2. α α α + ∆

26 Lemma 4.2. We have the following formulas:

1 Z ∞ 1 ζα = √ ρα(t+1)/t dt; π 0 t(t + 1) 1 Z ∞ 1 θα = √ ραt/(t+1) dt. π 0 t(t + 1)

In particular the maps ζα and θα are unital τ-symmetric, completely positive. Proof. Since we have that √ Z ∞ s s = √ dt, 0 t(t + s) for s > 0, we deduce Z ∞ ρα ζα = √ dt. 0 t(t + ρα) −1 However since ρα = α(α + ∆) we conclude

ρα α 1 = = ρα(t+1)/t. t + ρα α(t + 1) + t∆ t + 1

The formula for θα follows similarly.

1/2 Lemma 4.3 (Sauvageot [Sau99]). The maps ψt = exp(−t∆ ) are unital, completely positive for all t > 0.

1/2 1/2 Proof. We have that ∆ = limα→∞ ∆ ◦ ζα = limα→∞ α(id − θα), hence

−αt ψt = lim e exp(αtθα). α→∞

Since θα is completely positive so is exp(αtθα) and hence by taking a limit so is ψt. Note that we have D(∆1/2) = D(δ).

Lemma 4.4 (Ozawa-Popa [OP10]). Consider x, y ∈ D(δ) ∩ N, then

1/2 ∗ ∗ 1/2 1/2 ∗ 2 k∆ (x )y + x ∆ (y) − ∆ (x y)k2 ≤ 16kxk∞kδ(x)kkyk∞kδ(y)k.

Proof. Note that by Theorem 3.7 we have that x∗y ∈ D(∆1/2) = D(δ) for all x, y ∈ D(δ) ∩ N. For x, y ∈ D(δ) ∩ N we have the carr´edu champ:

Γ(x∗, y) := ∆1/2(x∗)y + x∗∆1/2(y) − ∆1/2(x∗y)

d = | (ψ (x∗y) − ψ (x∗)ψ (y)) dt t=0 t t t ψ (x∗y) − ψ (x∗)ψ (y) = lim t t t . t→0 t

27 Hence if we deﬁne a sesquilinear form on N ⊗ D(δ) by hb ⊗ y, a ⊗ xi = τ(a∗Γ(x∗, y)b) then this (n) form is non-negative deﬁnite (apply Kadison’s inequality to the maps ψt ). Thus we may use the Cauchy-Schwarz inequality to conclude

∗ 2 ∗ ∗ kΓ(x , y)k2 = sup |τ(a Γ(x , y)b)| ∗ ∗ kaa k2,kbb k2≤1

∗ ∗ ∗ ∗ ∗ ∗ ≤ sup τ(a Γ(x , x)a)τ(b Γ(y , y)b) ≤ kΓ(x , x)k2kΓ(y , y)k2. ∗ ∗ kaa k2,kbb k2≤1 1/2 ∗ ∗ Applying the triangle inequality together with the fact that k∆ (x x)k2 = kδ(x x)k then gives the result

Lemma 4.5. Let θ : N → N be a unital τ-symmetric completely positive map. If x, a ∈ N then

1/2 1/2 kθ(ax) − θ(a)θ(x)k2 ≤ 2kxk∞kak2 ka − θ(a)k2 .

Proof. If we consider the Stinespring dilation θ(x) = V π(x)V ∗ then we have

∗ ∗ ∗ ∗ kθ(ax) − θ(a)θ(x)k2 = kV π(x )(1 − VV )π(a )V k2

∗ 1/2 ∗ ∗ ∗ 1/2 ≤ kxk∞k(1 − VV ) π(a )V k2 = kxk∞τ(θ(aa ) − θ(a)θ(a )) ∗ ∗ 1/2 1/2 1/2 = kxk∞τ(aa − θ(a)θ(a )) ≤ 2kxk∞kak2 ka − θ(a)k2 .

Theorem 4.6 (Peterson [Pet09]). Let H be a Hilbert N-bimodule, let δ : L2(N, τ) → H be a −1/2 closable symmetric derivation. Denote by δ˜α(x) = α δ(ζα(x)). If x, a ∈ N then we have

˜ ˜ 1/2 1/2 ˜ 1/2 kζα(a)δα(x) − δα(ax))k ≤ 10kxk∞ kak2 kδα(a)k ;

˜ ˜ 1/2 1/2 ˜ 1/2 kδα(x)ζα(a) − δα(xa)k ≤ 10kxk∞ kak2 kδα(a)k . Proof. First note that for all y ∈ N we have

2 −1 ∗ kδ˜α(y)k = α h∆ ◦ ρα(y), yi = τ((y − ρα(y))y ).

2 2 ˜ Hence, since id − ρα ≥ 0 as an operator on L (N, τ) we have ky − ρα(y)k2 ≤ kδα(y)k ≤ kyk∞ky − ρα(y)k2, and kδ˜α(y)k ≤ kyk2. 2 Also note that as bounded√ √ operators on L (N, τ) it follows from functional calculus together 2 2 with the inequality 1 − t ≤ 1 − t for t ≥ 0 that we have (θα − ζα) ≤ (id − ζα) ≤ (id − ρα). Thus, for all y ∈ N we have

kθα(y) − ζα(y)k2 ≤ ky − ζα(y)k2 ≤ kδ˜α(y)k.

Suppose a, x ∈ N, then by using the product rule for δ we see that

−1/2 kζα(a)δ˜α(x) − α δ(ζα(a)ζα(x))k (7)

28 ˜ 1/2 ˜ 1/2 ≤ kxk∞kδα(a)k ≤ kxk∞kak∞ kδα(a)k . From Lemma 4.4 we have that

−1/2 1/2 −1/2 1/2 kα ∆ (ζα(a)ζα(x)) − α ζα(a)∆ (ζα(x))k2 (8)

1/2 ˜ 1/2 ≤ 5kxk∞kak∞ kδα(a)k . −1/2 1/2 Also recall that α ∆ ◦ ζα = id −θα and hence from Lemma 4.5 we have

−1/2 1/2 −1/2 1/2 kα ζα(a)∆ (ζα(x)) − α ∆ (ζα(ax))k2 (9)

= kζα(a)x − ζα(a)θα(x) − ax + θα(ax)k2

≤ ka − ζα(a)k2kxk∞ + kθα(a) − ζα(a)k2kxk∞ + kθα(a)θα(x) − θα(ax)k2 1/2 ˜ 1/2 ≤ 4kxk∞kak∞ kδα(a)k By considering the polar decomposition of δ = V ∆1/2, if we apply V to (8) and (9) above we have

−1/2 ˜ 1/2 ˜ 1/2 kα δ(ζα(a)ζα(x)) − δα(ax)k2 ≤ 9kxk∞kak∞ kδα(a)k (10)

Combining (7) and (10) then gives the result. The second inequality follows from the ﬁrst by symmetry.

The presence of the term ζα(a) in Theorem 4.6 can be undesirable but this is necessary since δ˜α(x) may not be a left or right bounded vector. This can be overcome however by changing our bimodule H slightly. Speciﬁcally, since ζα is unital, τ-symmetric, and completely positive it follows from Stinespring’s 0 theorem that there exists a normal pointed Hilbert N-bimodule (Hα, ξα) such that τ(ζα(x)y) = hxξαy, ξαi for all x, y ∈ N. Denote by Hα the Hilbert N-bimodule generated by vectors of the form 0 0 ξα ⊗N δ(x) ⊗N ξα for x ∈ D(δ). This is a subbimodule of Hα ⊗N H ⊗N Hα. 2 Denote by δα : L (N, τ) → Hα the bounded operator given by 1 δ (x) = √ ξ ⊗ δ(x) ⊗ ξ . α α α N N α

Note that δα(x) is well deﬁned since ξα is a left and right tracial vector. Moreover, by the deﬁnition of the relative tensor product we have that

haδα(x)b, δα(y)i = hζα(a)δ˜α(x)ζα(b), δ˜α(y)i, for all a, b, x, y ∈ N. In particular this shows that δα(x) is a left and right bounded vector for all x ∈ N and we can use the triangle inequality to deduce a similar approximate bimodularity property which we state now along with a number of other properties of δα which we have shown in this section.

29 Theorem 4.7. Let H be a normal Hilbert N-bimodule, let δ : L2(N, τ) → H be a closable symmetric 0 0 derivation. Let Hα ⊂ Hα ⊗N H ⊗N Hα and δα be deﬁned as above. Then for all a, x ∈ N we have 2 that δα : L (N, τ) → Hα is a contraction and

2 2 ∗ kx − ρα(x)k2 ≤ kδα(x)k = τ((x − ρα(x))x ) ≤ kxk2kx − ρα(x)k2;

ker(δα) ∩ N = ker(δ) ∩ N is a von neumann subalgebra of N; 2 α 7→ kδα(x)k is monotonically decreasing; 1/2 1/2 1/2 kδα(x)a − δα(xa)k ≤ 50kxk∞ kak2 kδα(a)k ; 1/2 1/2 1/2 kaδα(x) − δα(ax)k ≤ 50kxk∞ kak2 kδα(a)k .

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