aa r X i v : . [ m a t h . OA ] S e p POISSON BOUNDARIES OF II FACTORS
SAYAN DAS AND JESSE PETERSON
Abstract.
We introduce Poisson boundaries of II factors with respect to density oper-ators that give the traces. The Poisson boundary is a von Neumann algebra that containsthe II factor and is a particular example of the boundary of a unital completely positivemap as introduced by Izumi. Studying the inclusion of the II factor into its boundary wedevelop a number of notions, such as double ergodicity and entropy, that can be seen asnatural analogues of results regarding the Poisson boundaries introduced by Furstenberg.We use the techniques developed to answer a problem of Popa by showing that all finitefactors satisfy the MV-property. We also extend a result of Nevo by showing that property(T) factors give rise to an entropy gap. Introduction
Given a locally compact group G and a probability measure µ ∈ Prob( G ), the associated(left) random walk on G is the Markov chain on G whose transition probabilities are givenby the measures µ ∗ δ x . The Markov operator associated to this random walk is given by P µ ( f )( x ) = Z f ( gx ) dµ ( g ) , where f is a continuous function on G with compact support. The Markov operator extendsto a contraction on L ∞ ( G ), which is unital and (completely) positive. A function f ∈ L ∞ ( G )is µ -harmonic if P µ ( f ) = f . We let Har( G, µ ) denote the Banach space of µ -harmonicfunctions. The Furstenberg-Poisson boundary [Fur63b] of G with respect to µ is a certain G -probability space ( B, ζ ), such that we have a natural positivity preserving isometric G -equivariant identification of L ∞ ( B, ζ ) with Har(
G, µ ) via a Poisson transform.An actual construction of the Poisson boundary (
B, ζ ), which is often described as aquotient of the path space corresponding to the stationary σ -algebra, is less important tous here as its existence, and indeed, up to isomorphisms of G -spaces, it is the unique G -probability space such that L ∞ ( B, ζ ) is isomorphic, as any operator G -space, to Har( G, µ ).Under natural conditions on the measure µ , the boundary ( B, ζ ) possesses a number ofremarkable properties. It is an amenable G -space [Zim78], it is doubly ergodic with isometriccoefficients [Kai92] [GW16], and it is strongly asymptotically transitive [Jaw94, Jaw95]. Theboundary has therefore become a powerful tool for studying rigidity properties for groupsand their probability measure preserving actions [Mar75, Zim80, BS06, BM02, BF20].In light of the successful application of the Poisson boundary to rigidity properties ingroup theory, Alain Connes suggested (see [Jon00]) that developing a theory of the Poissonboundary in the setting of operator algebras would be the first step toward studying hisrigidity conjecture [Con82], which states that two property (T) icc groups have isomorphic J.P. was supported in part by NSF Grant DMS group von Neumann algebras if and only if the groups themselves are isomorphic. Furtherevidence for this can be seen by the significant role that Poisson boundaries play in [CP17,CP13, Pet15], where a related rigidity conjecture of Connes was investigated.Poisson boundaries can more generally be defined using any Markov operator associ-ated to a random walk. Markov operators are particular examples of normal unital com-pletely positive (u.c.p.) maps on von Neumann algebras, and motivated by defining Pois-son boundaries for discrete quantum groups, Izumi in [Izu02, Izu04] was able to define anon-commutative Poisson boundary associated to any normal u.c.p. map on a general vonNeumann algebra. Specifically, if M is a von Neumann algebra and φ : M → M is a normalu.c.p. map, then we let Har( φ ) = { x ∈ M | φ ( x ) = x } denote the space of φ -harmonic oper-ators. Izumi showed that there exists a (unique up to isomorphism) von Neumann algebra B φ such that, as operator systems, Har( φ ) and B φ can be identified via a Poisson transform P : B φ → Har( φ ). The existence of this boundary follows by showing that Har( φ ) can berealized as the range of a u.c.p. idempotent on M and then applying a theorem of Choiand Effros. Alternatively, the existence of the boundary follows by considering the minimaldilation of φ [Izu12]. We include in the appendix of this paper an elementary proof basedon this perspective.There is a well-known dictionary between many analytic notions in group theory andthose in von Neumann algebras. For example, states on B ( L ( M )) correspond to stateon ℓ ∞ Γ, normal Hilbert M -bimodules correspond to unitary representations, etc [Con76b,Section 2] [Con80]. This allows one to develop notions such as amenability, property (T), etc.in the setting of finite von Neumann algebras. While Izumi’s boundary gives a satisfactorynon-commutative analogue of the Poisson boundary associated to a general random walk,still missing is an appropriate notion of a non-commutative Poisson boundary analogous tothe group setting.The main goal of this article is to introduce a theory of Poisson boundaries for finite vonNeumann algebras that we believe will fill the role envisioned by Connes. If M is a finite vonNeumann algebra with a normal faithful trace τ , and if ϕ ∈ B ( L ( M, τ )) ∗ is a normal statesuch that ϕ | M = τ , then we will view ϕ as the distribution of a “non-commutative randomwalk” on M . To each distribution we associate a corresponding “convolution operator”,which is a normal u.c.p. map P ϕ : B ( L ( M, τ )) → B ( L ( M, τ )), such that M ⊂ Har( P ϕ ).We then define the Poisson boundary of M with respect to ϕ to be Izumi’s non-commutativeboundary B ϕ associated to P ϕ ; more precisely the boundary is really the inclusion of vonNeumann algebras M ⊂ B ϕ , together with the Poisson transform P : B ϕ → Har( P ϕ ).Poisson boundaries of groups give rise to natural Poisson boundaries of group von Neu-mann algebras. Indeed, as was already noticed by Izumi in [Izu12], if Γ is a group and µ ∈ Prob(Γ), then the non-commutative boundary of the u.c.p. map φ µ : B ( ℓ Γ) → B ( ℓ Γ)given by φ µ ( T ) = R ρ γ T ρ ∗ γ dµ ( γ ) is naturally isomorphic to the von Neumann crossed-product L ∞ ( B, ζ ) ⋊ Γ where (
B, ζ ) is the Poisson boundary of (
B, ζ ). Thus, many of theresults we obtain are not merely analogues, but are actually generalizations of results fromthe theory of random walks on groups.If M is a finite factor, then under natural conditions on the distribution ϕ , e.g., that its“support” should generate M , we show that the boundary B ϕ is amenable/injective (Propo-sition 2.3), and that the inclusion M ⊂ B ϕ is “ergodic”, i.e., M ′ ∩ B ϕ = C (Proposition 2.6).We use techniques of Foguel [Fog75] to obtain equivalent characterizations for when the OISSON BOUNDARIES OF II FACTORS 3 boundary is trivial (Theorem 2.9). The double ergodicity result of Kaimanovich [Kai92] ismore subtle, as unlike in the case for groups, there is no natural “diagonal” inclusion of M into B ϕ ⊗ B ϕ . There is however a natural notions of left and right convolution operators, sothat we may naturally associate with ϕ a second u.c.p. map P o ϕ which commutes with P ϕ .We may then show that bi-harmonic operators are constant, a result which is equivalent todouble ergodicity in the group setting. Theorem A (Theorem 3.1 below) . Let M be a finite factor and suppose ϕ is as above.Then we have Har( B ( L ( M, τ )) , P ϕ ) ∩ Har( B ( L ( M, τ )) , P o ϕ ) = C . Motivated by the question of determining whether or not L F ∞ is finitely generated,Popa studied in [Pop18] the class of separable II factors M that are tight, i.e., M containstwo hyperfinte subfactors L, R ⊂ M such that L and R op together generate B ( L ( M )).He conjectures in Conjecture 5.1 of [Pop18] that if a factor M has the property that allamplifications M t are singly generated then M is tight. He also notes that a tight factor M satisfies the MV-property, which states that for any operator T ∈ B ( L ( M )) the weakclosure of the convex hull of { u ( J vJ ) T ( J v ∗ J ) u ∗ | u, v ∈ U ( M ) } intersects the scalars. Popathen asks in Problem 7.4 of [Pop19a] and Problem 6.3 in [Pop19b] if free group factors, orperhaps all finite factors have the MV-property. As a consequence of double ergodicity weare able to answer Popa’s problem. Theorem B (Theorem 3.3 below) . All finite factors have the MV-property.
Other consequences of double ergodicity are that it allows us show vanishing cohomologyfor sub-bimodules of the Poisson boundary (Theorem 3.5), it allows us to generalize rigidityresults from [CP13] (Theorem 4.1), and it allows us to extend results of Bader and Shalom[BS06] identifying the Poisson boundary of a tensor product with the tensor product of thePoisson boundaries (Corollary 4.5).We also introduce analogues of Avez’s asymptotic entropy and Furstenberg’s µ -entropyin the setting of von Neumann algebras (see Section 5 for these definitions). We showthat non-triviality of the Poisson boundary is equivalent to vanishing Furstenberg entropy(Corollary 5.15). We also use entropy to extend a result Nevo [Nev03] to the setting of vonNeumann algebras, which shows that property (T) factors give rise to an “entropy gap”. Theorem C (Theorem 6.2 below) . Let M be a II factor with property (T) generated byunitaries u , . . . , u n . Define the state ϕ ∈ B ( L M ) ∗ by ϕ ( T ) = n P nk =1 h T ˆ u k , ˆ u k i . Thereexists c > such that if M ( A is an irreducible inclusion of von Neumann algebras and ζ ∈ A ∗ is any faithful normal state such that ζ | M = τ , then h ϕ ( M ⊂ A , ζ ) ≥ c . We end with an appendix where we construct Izumi’s boundary of a u.c.p. map. Ourapproach is elementary, and has the advantage that it applies for general C ∗ -algebras. Thislevel of generality has no doubt been known by experts, but we could not find this in thecurrent literature. 2. Boundaries
Hyperstates and bimodular u.c.p. maps.
Fix a tracial von Neumann algebra(
M, τ ), and suppose we have an embedding M ⊂ A where A is a C ∗ -algebra. We say SAYAN DAS AND JESSE PETERSON a state ϕ ∈ A ∗ is a τ -hyperstate (or just a hyperstate if τ is fixed) if it extends τ . Wedenote by S τ ( A ) the convex set of all hyperstates on A . To each a hyperstate ϕ we obtain anatural inclusion L ( M, τ ) ⊂ L ( A , ϕ ) induced from the map x ˆ1 x ϕ for x ∈ M . We let e M ∈ B ( L ( A , ϕ )) denote the orthogonal projection onto L ( M, τ ). We may then considerthe unital completely positive (u.c.p.) map P ϕ : A → B ( L ( M, τ )), defined by(1) P ϕ ( T ) = e M T e M , T ∈ A . Note that if x ∈ M ⊂ A then we have P ϕ ( x ) = x . We call the map P ϕ the Poisson transform(with respect to ϕ ) of the inclusion M ⊂ A .The following proposition is inspired from [Con76b, Section 2.2]. Proposition 2.1.
The correspondence ϕ
7→ P ϕ defined by (1) gives a bijective correspon-dence between hyperstates on M , and u.c.p., M -bimodular maps from A to B ( L ( M, τ )) .Moreover, if A is a von Neumann algebra, then P ϕ is normal if and only if ϕ is normal.Also, this corresondence is a homeomorphism where the space of hyperstates is endowedwith the weak ∗ -topology, and the space of u.c.p., M -bimodular maps with the topology ofpointwise weak operator topology convergence.Proof. First note that if ϕ is a hyperstate on A , then for all T ∈ A we have ϕ ( T ) = h T, ˆ1 i ϕ = hP ϕ ( T )ˆ1 , ˆ1 i τ . From this it follows that the correspondence ϕ
7→ P ϕ is one-to-one. To see that it is onto,suppose that P : A → B ( L ( M, τ )) is u.c.p. and M -bimodular. We define a state ϕ on A by ϕ ( T ) = hP ( T )ˆ1 , ˆ1 i τ . For all y ∈ M we then have ϕ ( y ) = hP ( y )ˆ1 , ˆ1 i τ = τ ( y ), hence ϕ isa hyperstate. Moreover, if y, z ∈ M , and T ∈ A then we have hP ϕ ( T ) y, z i τ = hP ϕ ( z ∗ T y )ˆ1 , ˆ1 i (2) = ϕ ( z ∗ T y ) = hP ( T ) y, z i τ , hence, P ϕ = P .It is also easy to check that P ϕ 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 ϕ α is a net of hyperstates. From (2) and the factthat u.c.p. maps are contractions in norm we see that P ϕ α converges in the pointwise weakoperator topology to P ϕ if ϕ α converges weak ∗ to ϕ . Conversely, setting y = z = 1 in(2) shows that if P ϕ α converges in the pointwise weak operator topology to P ϕ then ϕ α converges weak ∗ to ϕ . (cid:3) Considering the case A = B ( L ( M, τ )) we see that to each hyperstate ϕ on B ( L ( M, τ ))we obtain a u.c.p. M -bimodular map P ϕ on B ( L ( M, τ )). In particular, composing suchmaps gives a type of convolution operation on the space of hyperstates. More generally, if A is a C ∗ -algebra, with M ⊂ A , then for hyperstates ψ ∈ A ∗ , and ϕ ∈ B ( L ( M, τ )) ∗ wedefine the convolution ϕ ∗ ψ to be the unique hyperstate on A such that(3) P ϕ ∗ ψ = P ϕ ◦ P ψ . We say that ψ is ϕ -stationary if we have ϕ ∗ ψ = ψ , or equivalently, if P ψ maps into thespace of P ϕ -harmonic operatorsHar( P ϕ ) = Har( B ( L ( M, τ )) , P ϕ ) = { T ∈ B ( L ( M, τ )) | P ϕ ( T ) = T } . OISSON BOUNDARIES OF II FACTORS 5
Lemma 2.2.
For a fixed ψ ∈ S τ ( A ) the mapping S τ ( B ( L ( M, τ )) ∋ ϕ ϕ ∗ ψ ∈ S τ ( A ) is continuous in the weak ∗ -topology.Moreover, if ϕ ∈ B ( L ( N, τ )) ∗ is a fixed normal hyperstate, then the mapping S τ ( A ) ∋ ψ ϕ ∗ ψ ∈ S τ ( A ) is also weak ∗ -continuous.Proof. By Proposition 2.1 the correspondence ϕ
7→ P ϕ is a homeomorphism from the weak ∗ -topology to the topology of pointwise weak operator topology convergence, this lemma thenfollows easily from (3). (cid:3) Poisson boundaries of II factors. If ϕ ∈ S τ ( B ( L ( M, τ )) is a hyperstate then wedefine the Poisson boundary B ϕ of M with respect to ϕ to be the noncommutative Poissonboundary of the u.c.p. map P ϕ as defined by Izumi [Izu02], i.e., the Poisson boundary B ϕ isa C ∗ -algebra (a von Neumann algebra when ϕ is normal) that is isomorphic, as an operatorsystem, to the space of harmonic operators Har( B ( L ( M, τ )) , P ϕ ).The Poisson boundary contains M as a subalgebra, and the inclusion ( M ⊂ B ϕ ) isdetermined up to isomorphism by the property that there exists a completely positiveisometric surjection P : B ϕ → Har( B ( L ( M, τ )) , P ϕ ) which restricts to the identity map on M . We will always assume that P is fixed and we also call P the Poisson transform.Given any initial hyperstate ϕ ∈ S τ ( B ( L ( M, τ ))) we may consider the hyperstate givenby ϕ ◦ P on B ϕ . Of particular interest is the state η on B ϕ arising from the initial state x
7→ h x ˆ1 , ˆ1 i , which we call the stationary state on B ϕ . In this case it is easy to see that wehave P η = P , and hence ϕ ∗ η = η . Proposition 2.3.
Let ( M, τ ) be a tracial von Neumann algebra and fix a hyperstate ϕ on B ( L ( M, τ )) , then the Poisson boundary B ϕ is injective.Proof. If we take any accumulation point E of n N P Nn =1 P nϕ o N ∈ N in the topology of point-wise ultraweak convergence, then E : B ( L ( M, τ )) → Har( B ( L ( M, τ )) , P ϕ ) gives a condi-tional expectation. As B ϕ is isomorphic to Har( B ( L ( M, τ )) , P ϕ ) as an operator system itthen follows that B ϕ is injective [CE77, Section 3]. (cid:3) The trivial case is when ϕ e ( x ) = h x , i τ in which case we have that P ϕ e = id, andthe Poisson boundary is nothing but B ( L ( M, τ )). Note that ϕ e gives an identity withrespect to convolution. Also note that if ϕ ∈ B ( L ( M, τ )) ∗ is a hyperstate, then we have adescription of the space of harmonic operators as:Har( B ( L ( M, τ )) , P ϕ ) = { T ∈ B ( L ( M, τ )) | ϕ ( aT b ) = ϕ e ( aT b ) for all a, b ∈ M } . Since P ϕ is M -bimodular it follows that P ϕ ( M ′ ) ⊂ M ′ . We say that ϕ is regular if therestriction of P ϕ to M ′ preserves the canonical trace on M ′ , and we say that ϕ is generating if M is the largest ∗ -subalgebra of B ( L ( M, τ )) which is contained in Har( B ( L ( M, τ )) , P ϕ ).If ϕ is regular, then the conjugate of ϕ is given by ϕ ∗ ( T ) = ϕ ( J T ∗ J ), which is again ahyperstate. We’ll say that ϕ is symmetric if it is regular and we have ϕ ∗ = ϕ . SAYAN DAS AND JESSE PETERSON
Regular, generating, symmetric hyperstates are easy to find. Suppose (
M, τ ) is a sep-arable finite von Neumann algebra with a faithful normal trace τ . We consider the unitball ( M ) of M as a Polish space endowed with the strong operator topology, and supposewe have a σ -finite measure µ on ( M ) such that R x ∗ x dµ ( x ) = 1. We obtain a normalhyperstate as(4) ϕ ( T ) = Z h T c x ∗ , c x ∗ i dµ ( x )and using (2) we may explicitly compute the Poisson transform P ϕ on B ( L ( M, τ )) as P ϕ ( T ) = Z ( J x ∗ J ) T ( J xJ ) dµ ( x ) . Proposition 2.4.
Consider ϕ as given by (4), then(1) ϕ is generating if and only if the support of µ generates M as a von Neumannalgebra.(2) ϕ is regular if and only if R xx ∗ dµ ( x ) = 1 .(3) If ϕ is regular then P ϕ ∗ ( T ) = R ( J xJ ) T ( J x ∗ J ) dµ ( x ) and ϕ is symmetric if J ∗ µ = µ ,where J is the adjoint operation.Proof. If the support of µ generates von Neumann algebra M ⊂ M such that M = M ,then we have [ J xJ, e M ] = [ J x ∗ J, e M ] = 0 for each x in the support of µ . Hence, P ϕ ( T ) = R ( J xJ ) T ( J x ∗ J ) dµ ( x ) = T , for each T in the ∗ -algebra generated by M and e M . Therefore, ϕ is not generating. On the other hand, if T ∈ Har( B ( L ( M, τ )) , P ϕ ) is such that we alsohave T ∗ T, T T ∗ ∈ Har( B ( L ( M, τ )) , P ϕ ) then for each a ∈ M we have Z k (( J xJ ) T − T ( J xJ ))ˆ a k dµ ( x )= h ( T ∗ P ϕ (1) T − P ϕ ( T ∗ ) T − T ∗ P ϕ ( T ) + P ϕ ( T ∗ T ))ˆ a, ˆ a i = 0 , and by symmetry we also have R k (( J xJ ) T ∗ − T ∗ ( J xJ ))ˆ a k dµ ( x ) = 0. Hence, [ J xJ, T ] =[
J x ∗ J, T ] = 0 for µ -almost every x ∈ ( M ) . Therefore, if the support of µ generates M as a von Neumann algebra then we then have that T ∈ J M J ′ = M , showing that ϕ isgenerating.If y ∈ M then we have P ϕ ( J yJ ) = R J x ∗ yxJ dµ ( x ). Hence, we see that ϕ is regular ifand only if for all y ∈ M we have τ ( y ) = R τ ( x ∗ yx ) dµ ( x ) = R τ ( xx ∗ y ) dµ ( x ), which is ifand only if R xx ∗ dµ ( x ) = 1.If ϕ is regular then ϕ ∗ ( T ) = ϕ ( J T ∗ J ) = Z h J T ∗ J c x ∗ , c x ∗ i dµ ( x )= Z h ˆ x, T ∗ ˆ x i dµ ( x ) = Z h T c x ∗ , c x ∗ i dJ ∗ µ ( x ) . Therefore, if J ∗ µ = µ then ϕ is symmetric. (cid:3) The following lemma is well known, see, e.g., [FNW94], or Lemma 3.4 in [BJKW00]. Weinclude a proof for the convenience of the reader.
OISSON BOUNDARIES OF II FACTORS 7
Lemma 2.5.
Suppose A is a unital C ∗ -algebra with a faithful state ϕ . If P : A → A is au.c.p. map such that ϕ ◦ P = ϕ , then Har( A, P ) ⊂ A is a C ∗ -subalgebra.Proof. Har( A, P ) is clearly a self-adjoint closed subspace, thus we must show that Har( A, P )is an algebra. By the polarization identity it is enough to show that x ∗ x ∈ Har( A, P )whenever x ∈ Har( A, P ). Suppose x ∈ Har( A, P ). By Kadison’s indequality we have P ( x ∗ x ) − x ∗ x = P ( x ∗ x ) − P ( x ∗ ) P ( x ) ≥
0. Also, ϕ ( P ( x ∗ x ) − x ∗ x ) = 0 so that by faithfulnessof ϕ we have P ( x ∗ x ) = x ∗ x . (cid:3) Proposition 2.6.
Let M be a finite von Neumann algebra with a normal faithful trace τ .Let ϕ ∈ B ( L ( M, τ )) ∗ be a regular generating hyperstate, and let B ϕ be the correspondingPoisson boundary, then M ′ ∩ B ϕ = Z ( M ) . In particular, if ϕ is regular and normal and M is a factor then B ϕ is also a von Neumann factor.Proof. Let P : B ϕ → Har( B ( L ( M, τ )) , P ϕ ) denote the Poisson transform. If x ∈ M ′ ∩B ϕ , then P ( x ) ∈ M ′ ∩ B ( L ( M, τ )) =
J M J . Since ϕ is regular, P ϕ preserves the tracewhen restricted to J M J . Thus, Har( P ϕ , J M J ) is a von Neumann subalgebra of J M J byLemma 2.5, which must be Z ( M ) since ϕ is generating. Therefore, P ( x ) ∈ Har( P ϕ , J M J ) = Z ( M ),and hence x ∈ Z ( M ) since P is injective. (cid:3) If ϕ is a normal hyperstate in S τ ( B ( L ( M, τ ))), then P ϕ : B ( L ( M, τ )) → B ( L ( M, τ ))is a normal map, and hence the dual map P ∗ ϕ preserves the predual of B ( L ( M, τ )) whichwe identify with the space of trace-class operators.We let A ϕ ∈ B ( L ( M, τ )) denote the density operator associated with ϕ , i.e., A ϕ is theunique trace-class operator so that ϕ ( T ) = Tr( A ϕ T ) for all T ∈ B ( L ( M, τ )). Since ϕ ispositive we have that A ϕ is a positive operator. If P ˆ1 denotes the rank one orthogonalprojection onto C ˆ1, then we have ϕ ( T ) = hP ϕ ( T )ˆ1 , ˆ1 i = Tr( P ϕ ( T ) P ˆ1 ), and hence we seethat A ϕ = P ∗ ϕ ( P ˆ1 ). In particular we have that A ϕ ∗ n = ( P nϕ ) ∗ ( P ˆ1 ) for n ≥ Proposition 2.7.
Let ( M, τ ) be a tracial von Neumann algebra and let ϕ ∈ S τ ( B ( L ( M, τ ))) be a normal hyperstate, then there exists a τ -orthogonal family { z n } n which gives a partitionof the identity as P n z ∗ n z n so that P ϕ ( T ) = X n ( J z ∗ n J ) T ( J z n J ) for all T ∈ B ( L ( M, τ )) .Moreover, if { ˜ z m } m is a τ -orthogonal family which gives a partition of the identity as P n ˜ z ∗ n ˜ z n , then the map P m ( J ˜ z ∗ m J ) T ( J ˜ z m J ) agrees with P ϕ if and only if for each t > we have sp { z n | k z n k = t } = sp { ˜ z m | k ˜ z m k = t } . Proof.
Since A ϕ is a positive trace-class operator we may write A ϕ = P n a n P y n where a , a , . . . are positive and { y n } n is an orthonormal family with P y n denoting the rank oneprojection onto C y n . For T ∈ B ( L ( M, τ )) we then haveTr(
T A ϕ ) = X n a n h T y n , y n i . SAYAN DAS AND JESSE PETERSON
Taking T = x ∗ x ∈ M we have a n k xy n k ≤ Tr( x ∗ xA ϕ ) = k x k , so that y n ∈ M ⊂ L ( M, τ )for each n . Hence, for T ∈ B ( L ( M, τ )) we haveTr( P ϕ ( T ) P ˆ1 ) = Tr( T A ϕ ) = *X n a n ( J y n J ) T ( J y ∗ n J )ˆ1 , ˆ1 + = Tr X n a n ( J y n J ) T ( J y ∗ n J ) ! P ˆ1 ! . Since P ϕ is M -bimodular and since J y n J ∈ M ′ it follows that for all x, y ∈ M we haveTr( P ϕ ( T ) xP ˆ1 y ) = Tr X n a n ( J y n J ) T ( J y ∗ n J ) ! xP ˆ1 y ! . In particular, setting T = y = 1 we have τ ( x ) = X n a n τ ( y ∗ n y n x ) , which shows that P n a n y ∗ n y n = 1.Since the span of operators of the form xP ˆ1 y is dense in the space of trace-class operatorsit then follows that P ϕ ( T ) = P n a n ( J y n J ) T ( J y ∗ n J ) for all T ∈ B ( L ( M, τ )). Setting z n = √ a n y ∗ n then finishes the existence part of the proposition.Suppose now that { ˜ z m } m is a τ -orthogonal family which gives a partition of the identity1 = P n ˜ z ∗ n ˜ z n , and set ˜ ϕ ( T ) = Tr(( P n ( J ˜ z ∗ n J ) T ( J ˜ z n J )) P ˆ1 ). Then, the density matrixcorresponding to ˜ ϕ is P n ˜ z ∗ n P ˆ1 ˜ z n . Since { ˜ z n } n forms a τ -orthogonal family it then followseasily that ˜ z ∗ n is an eigenvector for A ˜ ϕ , and the corresponding eigenvalue is k z ∗ n k = k z n k .Since { z n } n above was constructed using any orthonormal basis of eigenvectors from A ϕ the rest of the proposition then follows easily. (cid:3) We say that the form P ϕ ( T ) = P n ( J z ∗ n J ) T ( J z n J ) (resp. ϕ ( T ) = P n h T b z ∗ n , b z ∗ n i ) is astandard form for P ϕ (resp. ϕ ). It follows from Proposition 2.4 that ϕ is generating if andonly if { z n } generates M as a von Neumann algebra. We say that ϕ is strongly generatingif the unital algebra (rather than the unital ∗ -algebra) generated by { z n } is already weaklydense in M . This is the case, for example, if ϕ is generating and symmetric, since thenwe have that { z n } = { z ∗ n } , and hence the unital algebra generated by { z n } is already a ∗ -algebra. Proposition 2.8.
Let ( M, τ ) be a tracial von Neumann algebra and suppose ϕ is a normalstrongly generating hyperstate, then the stationary state ζ gives a normal faithful state onthe Poisson boundary B ϕ such that ζ | M = τ .Proof. By considering the Poisson transform, it suffices to show that ϕ is normal and faithfulon the operator system Har( P ϕ ). Note that here the stationary state is a vector state andhence normality follows. To see that the state is faithful fix T ∈ Har( P ϕ ), with T ≥ h T ˆ1 , ˆ1 i = 0. Let P ϕ ( S ) = P n ( J z ∗ n J ) S ( J z n J ) be the standard form of P ϕ . Since T ∈ Har( P ϕ ), we have that P kϕ ( T ) = T , for each k ∈ N . Expanding the standard form gives0 = h T ˆ1 , ˆ1 i = h P kϕ ( T )ˆ1 , ˆ1 i = X n ,n ,...,n k h T z n z n · · · z n k ˆ1 , z n z n · · · z n k ˆ1 i . OISSON BOUNDARIES OF II FACTORS 9
We than have T ˆ m = 0 for all m in the unital algebra generated by { z n } , and as ϕ is stronglygenerating it then follows that T = 0. (cid:3) We end this section by giving a condition for the boundary to be trivial. We denote thespace of trace-class operators on L ( M, τ ) by TC( L ( M, τ )). We also denote the trace-classnorm on TC( L ( M, τ )) by k · k TC . Theorem 2.9.
Let ( M, τ ) be a tracial von Neumann algebra and let ψ be a normal hy-perstate. set ϕ = ψ + h· ˆ1 , ˆ1 i and let A n ∈ TC( L ( M, τ )) denote the density matrixcorresponding to the normal, u.c.p. M -bimodular map P nϕ . Then the following conditionsare equivalent(1) For all x ∈ M we have k xA n − A n x k TC → .(2) For all x ∈ M we have xA n − A n x → weakly.(3) Har( P ϕ ) = M Proof.
The first condition trivially implies the second. To see that the second implies thethird suppose for each x ∈ M we have xA n − A n x → n → ∞ . Let T ∈ Har( P ϕ ).Let x, a, b ∈ M . Then we have: |h ( T J xJ − J xJ T ) a ˆ1 , b ˆ1 i| = |h ( b ∗ T ax ∗ − x ∗ b ∗ T a )ˆ1 , ˆ1 i| = |hP nϕ ( b ∗ T ax ∗ − x ∗ b ∗ T a )ˆ1 , ˆ1 i| = | Tr(A n (b ∗ Tax ∗ − x ∗ b ∗ Ta)) | = | Tr((x ∗ A n − A n x ∗ )b ∗ Ta) | → . Hence T ∈ J M J ′ = M .To see that the third condition implies the first we adapt the approach of Foguel from[Fog75]. Suppose Har( P ϕ ) = M . Set A = { A ∈ TC( L ( M, τ )) | k ( P nϕ ) ∗ ( A ) k TC → } .Note that since ( P nϕ ) ∗ is a contraction in the trace-class norm we have that A is a closedsubspace.Since ϕ = ψ + h· ˆ1 , ˆ1 i we have P ∗ ϕ = id + P ∗ ψ and we compute( P nϕ ) ∗ (id − P ∗ ϕ ) = 2 − ( n +1) n X k =0 (cid:18) nk (cid:19) ( P kψ ) ∗ ! (id − P ∗ ψ )= 2 − ( n +1) n X k =1 (cid:18)(cid:18) nk − (cid:19) − (cid:18) nk (cid:19)(cid:19) P ∗ ψ . We have lim n →∞ − ( n +1) P nk =1 (cid:16)(cid:0) nk − (cid:1) − (cid:0) nk (cid:1)(cid:17) = 0 (see (1.8) in [OS70]) hence k ( P nϕ ) ∗ ( P ˆ1 −P ∗ ϕ ( P ˆ1 )) k TC →
0. Thus P ˆ1 − P ∗ ϕ ( P ˆ1 ) ∈ A .Since P ∗ ϕ is M -bimodular we then have that aP ˆ1 b − P ∗ ϕ ( aP ˆ1 b ) ∈ A for each a, b ∈ M and hence B − P ∗ ϕ ( B ) ∈ A for all B ∈ TC( L ( M, τ )). If T ∈ B ( L ( M, τ )) is such thatTr( AT ) = 0 for all A ∈ A , then for all B ∈ TC( L ( M, τ )) we have h B − P ∗ ϕ ( B ) , T i = 0so that T ∈ Har( P ϕ ) = M . By the Hahn-Banach theorem it then follows that A ∈ A whenever Tr( Ax ) = 0 for all x ∈ M . In particular, we have xP ˆ1 − P ˆ1 x ∈ A for all x ∈ M ,which is equivalent to the fact that k xA n − A n x k TC → x ∈ M . (cid:3) Biharmonic operators If ϕ ∈ S τ ( B ( L ( M, τ ))) is regular and normal then we define P o ϕ to be the u.c.p. mapgiven by P o ϕ = Ad( J ) ◦ P ϕ ∗ ◦ Ad( J ). Note that P o ϕ and P η commute for any normalhyperstate η . Indeed, if we have standard forms P ϕ ( T ) = P n ( J z ∗ n J ) T ( J z n J ) and P η ( T ) = P m ( J y ∗ m J ) T ( J y m J ) then by Proposition 2.4 we have P o ϕ ( T ) = P n z n T z ∗ n and hence P o ϕ ◦ P η ( T ) = P η ◦ P o ϕ ( T ) = X n,m z n ( J y ∗ m J ) T ( J y m J ) z ∗ n . The following is a noncommutative analogue of double ergodicity which was establishedin [Kai92].
Theorem 3.1.
Let ( M, τ ) be a tracial von Neumann algebra and let ϕ be a normal regularstrongly generating hyperstate. Then Har( B ( L ( M, τ )) , P ϕ ) ∩ Har( B ( L ( M, τ )) , P o ϕ ) = Z ( M ) . Proof.
We fix a standard form P ϕ ( T ) = P n ( J z ∗ n J ) T ( J z n J ), so that we also have P o ϕ ( T ) = P m z m T z ∗ m . We identify the Poisson boundary B ϕ with Har( B ( L ( M, τ ))), and let ζ denotethe stationary state on B ϕ , which is faithful by Proposition 2.8. For T ∈ B ϕ we have ζ ( P o ϕ ( T )) = hP o ϕ ( T )ˆ1 , ˆ1 i = hP ϕ ( T )ˆ1 , ˆ1 i = ζ ( P ϕ ( T )) = ζ ( T ) . By Lemma 2.5 we then have that B = Har( B ϕ , P o |B ϕ ) is a von Neumann subalgebra of B ϕ . If p ∈ B is a projection and ξ ∈ L ( B ϕ , ζ ) then X n k pz ∗ n p ⊥ ξ k = X n h z n pz ∗ n p ⊥ ξ, p ⊥ ξ i = 0 . We must therefore have k pz ∗ n p ⊥ ξ k = 0 for each n , and hence pz ∗ n = pz ∗ n p , for each n .Repeating this argument with roles of p and p ⊥ reversed shows that z ∗ n p = pz ∗ n p , so that p ∈ M ′ ∩ B ϕ . Since p was an arbitrary projection we then have B ⊂ M ′ ∩ B ϕ and byProposition 2.6 we have B = Z ( M ). (cid:3) The previous result allows us to give an analogue of the classical Choquet-Deny theorem[CD60], which states that if Γ is an abelian group and µ ∈ Prob(Γ) has support generatingΓ then every bounded µ -harmonic function is constant. Corollary 3.2 (The Choquet-Deny theorem) . Suppose M is an abelian von Neumannalgebra and ϕ is a normal regular strongly generating hyperstate, then Har( B ( L ( M, τ )) , P ϕ ) = Z ( M ) = M. We will now describe how Theorem 3.1 leads to a positive answer of a recent question byPopa [Pop19a, Problem 7.4] [Pop19b, Problem 6.3].
Theorem 3.3.
Let M be a finite von Neumann algebra with a normal faithful trace τ and let G ⊂ U ( M ) be a group which generates M as a von Neumann algebra. Then for any operator T ∈ B ( L ( M, τ )) the weak closure of the convex hull of { u ( J vJ ) T ( J v ∗ J ) u ∗ | u, v ∈ G} intersects Z ( M ) . OISSON BOUNDARIES OF II FACTORS 11
Proof.
We first consider the case when G is countable. Let µ ∈ Prob( G ) by symmet-ric with full support and define a normal regular symmetric generating hyperstate ϕ by ϕ ( T ) = R h T ˆ u, ˆ u i dµ ( u ). The corresponding Poisson transform is then given by P ϕ ( T ) = R ( J uJ ) T ( J u ∗ J ) dµ ( u ), and we may also compute P o ϕ as P o ϕ ( T ) = R u ∗ T u dµ ( u ).Fix T ∈ B ( L ( M, τ )) and let C = co wk { u ( J vJ ) T ( J v ∗ J ) u ∗ | u, v ∈ G} . Then C is preservedby both P ϕ and P o ϕ and hence C is preserved by any point-ultraweak limit points E and E o of n N P Nn =1 P nϕ o ∞ N =1 and n N P Nn =1 ( P o ϕ ) n o ∞ N =1 respectively. Since P ϕ and P o ϕ commutewe have that E and E o commute. Moreover, as k N P Nn =1 P nϕ − N P Nn =1 P n +1 ϕ k ≤ /N itfollows that E : B ( L ( M, τ )) → Har( P ϕ ) and similarly E o : B ( L ( M, τ )) → Har( P o ϕ ). ByTheorem 3.1 we then have E o ◦ E : B ( L ( M, τ )) → Z ( M ). Hence E o ◦ E ( T ) ∈ C ∩ Z ( M ) . In the general case, if
G < G is a countable subgroup, then let N ⊂ M be the vonNeumann subalgebra generated by G and let e N : L ( M, τ ) → L ( N, τ ) be the orthogonalprojection. If we define ϕ as above and set T G = E o ◦ E ( T ) then we have T G ∈ C , e N T G e N = E o ◦ E ( e N T e N ) and viewing eT e as an operator in B ( L ( N, τ )) we may applyTheorem 3.1 as above to conclude that eT e ∈ Z ( N ) ⊂ B ( L ( N, τ )). If we consider thenet { T G } G ⊂ B ( L ( M, τ )) where G varies over all countable subgroups of G , ordered byinclusion, then letting T be any weak limit point of this net we have that T ∈ C and if G < G is a countable subgroup generating a von Neumann subalgebra N ⊂ M , and x ∈ N then we have e N [ x, T ] e N = [ x, e N T e N ] = 0. Since G generates M it therefore follows that T ∈ Z ( M ). (cid:3) The derivation problem in von Neumann algebras asks if for each von Neumann algebra M there exists a constant k > T ∈ B ( L M ) we have the distance inequalitydist( T, M ′ ) ≤ k k δ T | M k , where δ T is the derivation given by δ T ( x ) = [ x, T ]. While we arenot able to shed light on this problem, we are able to obtain the following variant, whichmay be of interest. Corollary 3.4.
Let M be a finite von Neumann algebra, and suppose T ∈ B ( L ( M )) , then dist( T, Z ( M )) ≤ k δ T | M ′ k + k δ T | M k . Proof.
This follows from the previous theorem since every point S ∈ { u ( J vJ ) T ( J v ∗ J ) u ∗ | u, v ∈ U ( M ) } satisfies dist( T, S ) ≤ k δ T | M ′ k + k δ T | M k . (cid:3) As another application of Theorem 3.1 we use Christensen’s Theorem [Chr82, Theorem5.3] to establish the following vanishing cohomology result; the case when C = M is thecelebrated Kadison-Sakai theorem [Kad66, Sak66]. Theorem 3.5.
Let ( M, τ ) be a tracial von Neumann algebra and let ϕ be a normal regularstrongly generating hyperstate, suppose C ⊂ B ϕ is a weakly closed M -bimodule. If δ : M → C is a norm continuous derivation then there exists c ∈ C so that δ ( x ) = [ x, c ] for x ∈ M .Moreover, if ϕ has the form ϕ ( T ) = R h T c u ∗ , c u ∗ i dµ ( u ) for some probability measure µ ∈U ( M ) , then c may be chosen so that k c k ≤ k δ k .Proof. Identifying C with its image under the Poisson transform we will view C as anoperator system in Har( P ϕ ) ⊂ B ( L ( M, τ )). Since L ( M, τ ) has a cyclic vector for M , Christensen’s Theorem [Chr82, Theorem 5.3] shows that δ ( m ) = mT − T m for some T ∈ B ( L ( M, τ )). Taking the conditional expectation onto Har( P ϕ ), we may assume T ∈ Har( P ϕ ).We suppose ϕ is given in standard form ϕ ( T ) = P n h T b z ∗ n , b z ∗ n i . Note that z m δ ( z ∗ m ) ∈ C ,so that T − P o ϕ ( T ) = X m z m z ∗ m T − X m z m T z ∗ m = X m z m δ ( z ∗ m ) ∈ C As P o ϕ leaves C invariant (since C is an M -bimodule), by induction we get that T − ( P o ϕ ) n ( T ) ∈ C for all n ≥
1, and hence for N ≥ T − N N X n =1 ( P o ϕ ) n ( T ) ∈ C If z is a weak limit point of (cid:26) N P Nn =1 ( P o ϕ ) n ( T ) (cid:27) then z ∈ Har( P o ϕ ) ∩ Har( P ϕ ) and so byTheorem 3.1 we have z ∈ Z ( M ). Thus, T − z ∈ C implements the derivation.For the moreover part, note that if ϕ has the form ϕ ( T ) = R h T c u ∗ , c u ∗ i dµ ( u ) for someprobability measure µ ∈ U ( M ) then k T − z k ≤ sup N k T − N N X n =1 ( P o ϕ ) n ( T ) k≤ sup n k T − ( P o ϕ ) n ( T ) k = sup n k Z uδ ( u ∗ ) dµ n k ≤ k δ k , where µ n denotes the push forward of µ × µ × · · · µ ∈ Prob( U ( M ) n ) under the multiplicationmap.Hence c = T − z implements δ with k c k ≤ k δ k . (cid:3) We remark that for a general hyperstate ϕ , in the proof of the previous theorem we stillhave k T − z k ≤ k δ k cb . So that in general we may find c ∈ C with k c k ≤ k δ k cb .4. Rigidity for u.c.p. maps on boundaries
The main result in this section is Theorem 4.1, where we generalize [CP13, Theorem 3.2].We mention several consequences, including a noncommutative version of [BS06, Corollary3.2], which describes the Poisson boundary of a tensor product as the tensor product ofPoisson boundaries.
Theorem 4.1.
Let ( M, τ ) be a tracial von Neumann algebra, let ϕ be a normal regularstrongly generating hyperstate, and let B = B ϕ denote the corresponding boundary. Supposewe have a weakly closed operator system C such M ⊂ C ⊂ B . Let Ψ :
C → B be a normalu.c.p. map such that Ψ | M = id . Then Ψ = id . OISSON BOUNDARIES OF II FACTORS 13
Proof.
By identifying C with its image under the Poisson transform we may assume that C is a weakly closed M -subbimodule of Har( P ϕ ) and Ψ : C →
Har( P ϕ ) is a normal u.c.p. mapsuch that Ψ | M = id. Note that for T ∈ C we have, h Ψ( T )ˆ1 , ˆ1 i = hP ϕ (Ψ( T ))ˆ1 , ˆ1 i = hP o ϕ (Ψ( T ))ˆ1 , ˆ1 i = X n h z n Ψ( T ) z ∗ n ˆ1 , ˆ1 i = h Ψ( P o ϕ ( T ))ˆ1 , ˆ1 i , where the last equality follows from the fact that Ψ is normal and M -bimodular. Now, h Ψ( P o ϕ ( T ))ˆ1 , ˆ1 i = h Ψ( T )ˆ1 , ˆ1 i for all T ∈ C immediately implies that * Ψ N N X n =1 ( P o ϕ ) n ( T ) ! ˆ1 , ˆ1 + = h Ψ( T )ˆ1 , ˆ1 i for all T ∈ C . Let z be a weak operator topology limit point of N P Nn =1 ( P o ϕ ) n ( T ). Then, z ∈ Z ( M ) byTheorem 3.1, so that Ψ( z ) = z . We then have h Ψ( T )ˆ1 , ˆ1 i = h z ˆ1 , ˆ1 i = h T ˆ1 , ˆ1 i where the last equality follows because z is independent of Ψ. Now, let a, b ∈ M , and T ∈ C .Then, we have that b ∗ T a ∈ C , and hence by above computation, we get h Ψ( T ) a ˆ1 , b ˆ1 i = h Ψ( b ∗ T a )ˆ1 , ˆ1 i = h b ∗ T a ˆ1 , ˆ1 i = h T a ˆ1 , b ˆ1 i . Thus Ψ( T ) = T (cid:3) Corollary 4.2.
Let M be a finite von Neumann algebra with a normal faithful trace τ , andlet ϕ be a normal regular strongly generating hyperstate. Then, M is a maximal type II factor inside B ϕ . Proof.
Suppose N ⊂ B ϕ is a type II factor containing M . Then there exists a normalconditional expectation E : N → M . Hence, by Theorem 4.1, E ( x ) = x for all x ∈ N , andhence N = M . (cid:3) Corollary 4.3.
Let M be a II factor, and let ϕ be a normal regular strongly generatinghyperstate. Then, either B ϕ = M , or else B ϕ is a type III factor, such that the stationarystate ζ is normal and faithful.Proof. Note that the stationary state is normal and faithful by Proposition 2.8, and B ϕ isa factor by Proposition 2.6. Suppose B ϕ is not a type III factor, then B ϕ has a semi-finitenormal faithful trace Tr. As before, let P denote the Poisson transform, and let ζ be thenormal state on B ϕ defined by ζ ( b ) = hP ( b )ˆ1 , ˆ1 i . Fix 0 ≤ T ∈ B ϕ with T r ( T ) < ∞ .Note that T r ( P o ϕ ( T )) = T r ( T ), as P o ϕ ( T ) = P n z n T z ∗ n and P n z ∗ n z n = 1. Let z be aweak operator topology limit point of N P Nn =1 ( P o ϕ ) n ( T ). Then by Theorem 3.1 we have z ∈ Z ( M ) = C and arguing as in the proof of Theorem 4.1 we have ζ ( T ) = z = Tr( z ). Thisthen shows that ζ (and also T r ) is a normal tracial state. Hence B ϕ is a type II factor.and by Corollary 4.2 we have that B ϕ = M . (cid:3) Theorem 4.4.
Suppose for each i ∈ { , } , M i is a finite von Neumann algebra withnormal faithful trace τ i . Let ϕ i be a normal regular strongly generating hyperstate for M i on B ( L ( M i , τ i )) . Then, Har( P ϕ ⊗ P ϕ ) = Har( P ϕ ) ⊗ Har( P ϕ ) . Proof.
We clearly have Har( P ϕ ) ⊗ Har( P ϕ ) ⊂ Har( P ϕ ⊗ P ϕ ) so we only need to showthe reverse inclusion. Note that( P ϕ ⊗ id) ◦ ( P ϕ ⊗ P ϕ ) = ( P ϕ ⊗ P ϕ ) ◦ ( P ϕ ⊗ id) , hence ( P ϕ ⊗ id) | Har( P ϕ ⊗P ϕ ) gives a normal ucp map which restricts to the identity on M ⊗ M . By Theorem 4.1 we have that ( P ϕ ⊗ id) | Har( P ϕ ⊗P ϕ ) is the identity map andhence Har( P ϕ ⊗ P ϕ ) ⊂ Har( P ϕ ⊗ id) = Har( P ϕ ) ⊗ B ( L M ) . We similarly have Har( P ϕ ⊗ P ϕ ) ⊂ B ( L M ) ⊗ Har( P ϕ ) . Since Har( P ϕ ) is injective it is semidiscrete [Con76a], and hence has property S σ ofKraus [Kra83, Theorem 1.9]. We then haveHar( P ϕ ⊗ ϕ ) ⊂ (Har( P ϕ ) ⊗ B ( L M )) ∩ ( B ( L M ) ⊗ Har( P ϕ )) ⊂ Har( P ϕ ) ⊗ Har( P ϕ ) . (cid:3) Corollary 4.5.
Suppose for each i ∈ { , } , M i is a finite von Neumann algebra withnormal faithful trace τ i . Let ϕ i be a normal regular strongly generating hyperstate for M i on B ( L ( M i , τ i )) . Then, the identity map on M ⊗ M uniquely extends to a ∗ -isomorphismbetween B ϕ ⊗ ϕ and B ϕ ⊗ B ϕ . Entropy
In this section we introduce noncommutative analogues of Avez’s asymptotic entropy[Ave72], and Furstenberg entropy [Fur63a, Section 8].5.1.
Asymptotic entropy.
Let M be a tracial von Neumann algebra with a faithful nor-mal tracial state τ . For a normal hyperstate ϕ ∈ S τ ( B ( L ( M, τ ))) we define the entropy of ϕ , denoted by H ( ϕ ), to be the von Neumann entropy of the corresponding density matrix A ϕ : H ( ϕ ) = − Tr( A ϕ log( A ϕ )) . If we have a standard form ϕ ( T ) = P n h T b z ∗ n , b z ∗ n i then we may compute this explicitly as H ( ϕ ) = − X n k z n k log( k z n k ) . Theorem 5.1. If ϕ and ψ are two normal hyperstates with ψ regular, then H ( ϕ ∗ ψ ) ≤ H ( ϕ ) + H ( ψ ) OISSON BOUNDARIES OF II FACTORS 15
Proof.
Let A ϕ and A ψ be the corresponding density operators and P ϕ and P ψ be thecorresponding u.c.p. M -bimodular maps. Let { a i } i ∈ I and { c j } j ∈ J be τ orthogonal families,as given by Proposition 2.7, such that A ϕ = P i µ i P ˆ a i and A ψ = P j ν j P ˆ c j . Let b i = J a i J and d i = J c i J so that P ϕ ( T ) = X i µ i b i T b ∗ i and P ψ ( T ) = X j ν j d j T d ∗ j . Since ψ is regular we have that P i ν i d ∗ i d i = P i ν i d i d ∗ i = 1. Since ϕ is a hyperstate we havethat P i µ i b i b ∗ i = 1. Now, H ( A ϕ ∗ ψ ) = − X i,j T r [ µ i ν j b ∗ i d ∗ j P ˆ1 d j b i log( A ϕ ∗ ψ )] . and b ∗ i d ∗ j P ˆ1 d j b i = τ ( b i b ∗ i d ∗ j d j ) P d b ∗ i d ∗ j , so that for each k, ℓ we have A ϕ ∗ ψ = X i,j µ i ν j b ∗ i d ∗ j P ˆ1 d j b i ≥ µ k ν ℓ τ ( b k b ∗ k d ∗ ℓ d ℓ ) P d b ∗ k d ∗ ℓ . As log is operator monotone, for each k, ℓ we then have − log( A ϕ ∗ ψ ) = − log( X i,j µ i ν j b ∗ i d ∗ j P ˆ1 d j b i ) ≤ − log(( µ k ν ℓ τ ( b k b ∗ k d ∗ ℓ d ℓ )) P d b ∗ k d ∗ ℓ ) . Hence, H ( A ϕ ∗ ψ ) ≤ − X i,j T r [ µ i ν j τ ( b i b ∗ i d ∗ j d j ) P d b ∗ i d ∗ j log( µ i ν j τ ( b i b ∗ i d ∗ j d j ) P d b ∗ i d ∗ j )]= − X i,j T r [ µ i ν j τ ( b i b ∗ i d ∗ j d j ) P d b ∗ i d ∗ j log( µ i ν j τ ( b i b ∗ i d ∗ j d j ))] − X i,j T r [ µ i ν j τ ( b i b ∗ i d ∗ j d j ) P d b ∗ i d ∗ j log( P d b ∗ i d ∗ j )]= − X i,j µ i ν j τ ( b i b ∗ i d ∗ j d j ) log( µ i ν j τ ( b i b ∗ i d ∗ j d j )) . Now define m on I × J by m ( i, j ) = µ i ν j τ ( b i b ∗ i d ∗ j d j ). Note that X i m ( i, j ) = ν j τ ( X i µ i b i b ∗ i d ∗ j d j ) = ν j τ ( d ∗ j d j ) = ν j and X j m ( i, j ) = µ i τ ( X i ν j b i b ∗ i d ∗ j d j ) = µ i τ ( b i b ∗ i ) = µ i . To finish the proof it then suffices to show H ( m ) = − X i,j m ( i, j ) log( m ( i, j )) ≤ H ( µ ) + H ( ν ) . Note that H ( m ) = − X i,j m ( i, j ) log( m ( i, j ))= − X i,j µ i ν j τ ( b i b ∗ i d ∗ j d j ) log( µ i τ ( b i b ∗ i d ∗ j d j )) − X i,j µ i ν j τ ( b i b ∗ i d ∗ j d j ) log( ν j )= − X i,j µ i ν j τ ( b i b ∗ i d ∗ j d j ) log( µ i ) − X i,j µ i ν j τ ( b i b ∗ i d ∗ j d j ) log( ν j ) − X i,j µ i ν j τ ( b i b ∗ i d ∗ j d j ) log( τ ( b i b ∗ i d ∗ j d j )) . In the last equality above, the first summation is H ( µ ), since summing over j we get − X i µ i τ ( b i b ∗ i ) log( µ i ) = − X i µ i log( µ i ) , while the second summation is H ( ν ). Hence, all that remains is to show: X i,j µ i ν j τ ( b i b ∗ i d ∗ j d j ) log( τ ( b i b ∗ i d ∗ j d j )) ≥ . Let η ( x ) = − x log( x ) for x ∈ [0 , η is concave, and so η ( P i α i x i ) ≥ P i α i η ( x i )whenever α i ≥ P i α i = 1. So, − X i,j µ i ν j τ ( b i b ∗ i d ∗ j d j ) log( τ ( b i b ∗ i d ∗ j d j )) = X i,j µ i ν j η ( τ ( b i b ∗ i d ∗ j d j ))= X i µ i ( X j ν j η ( τ ( b i b ∗ i d ∗ j d j ))) ≤ X i µ i η ( X j ν j τ ( b i b ∗ i d ∗ j d j ))= X i µ i η ( τ ( b i b ∗ i )) = 0 (cid:3) Corollary 5.2. If ϕ is a normal regular hyperstate, then the limit lim n →∞ H ( ϕ ∗ n ) n exits.Proof. The sequence { H ( ϕ ∗ n ) } is subadditive by Theorem 5.1 and hence the limit exists. (cid:3) The asymptotic entropy h ( ϕ ) of a normal regular hyperstate ϕ is defined to be thelimit h ( ϕ ) = lim n →∞ H ( ϕ ∗ n ) n . OISSON BOUNDARIES OF II FACTORS 17
A Furstenberg type entropy.
Suppose G is a Polish group and µ ∈ Prob( G ). Givena quasi-invariant action G y a ( X, ν ) the corresponding Furstenberg entropy (or µ -entropy)is defined [Fur63a, Section 8] to be h µ ( a, ν ) = − Z Z log (cid:18) dg − νdν ( x ) (cid:19) dν ( x ) dµ ( g ) . If we consider the measure space ( G × X, ν × µ ) then we have a non-singular map π : G × X → G × X given by π ( g, x ) = ( g, g − x ), whose Radon-Nikodym derivative is given by dπ ( µ × ν ) d ( µ × ν ) ( x, g ) = dg − νdν ( x ) . We may thus rewrite the µ -entropy as a relative entropy h µ ( a, ν ) = − Z Z log (cid:18) dπ ( ν × µ ) d ( ν × µ ) ( g, x ) (cid:19) d ( ν × µ ) = S (( ν × µ ) | π ( ν × µ )) . Let (
M, τ ) be a tracial von Neumann algebra, ϕ a normal hyperstate for M , and A a C ∗ -algebra, such that M ⊆ A . Let ζ ∈ S τ ( A ) be a faithful hyperstate. Let ∆ ζ : L ( A , ζ ) → L ( A , ζ ) be the modular operator corresponding to ζ , and consider the spectraldecomposition ∆ ζ = R ∞ λ dE ( λ ).Since ζ | M = τ , we have a natural inclusion of L ( M, τ ) in L ( A , ζ ). Let e denote theorthogonal projection from L ( A , ζ ) to L ( M, τ ). The entropy of the inclusion (
M, τ ) ⊂ ( A , ζ ) with respect to ϕ is defined to be h ϕ ( M ⊂ A , ζ ) = − Z log( λ ) dϕ ( eE ( λ ) e ) . The next example shows that h ϕ ( M ⊂ A , ζ ) can be considered as a generalization of theFurstenberg entropy. Example 5.3.
If Γ is a discrete group, µ ∈ Prob(Γ) and Γ y a ( X, ν ) is a quasi-invariantaction, then we may consider the state ϕ on B ( ℓ Γ) given by ϕ ( T ) = R h T δ γ , δ γ i dµ ( γ ), andwe may consider the state ζ on L ∞ ( X, ν ) ⋊ Γ ⊂ B ( ℓ Γ ⊗ L ( X, ν )) given by ζ (cid:16)P γ ∈ Γ a γ u γ (cid:17) = R a e dν . Note that we have that in this case we may compute ϕ ∗ ζ (cid:16)P γ ∈ Γ a γ u γ (cid:17) = R a e dµ ∗ ν .The modular operator ∆ ζ is then affiliated to the von Neumann algebra ℓ ∞ Γ ⊗ L ∞ ( X, ν ),and we may compute this directly as∆ ζ ( γ, x ) = dγ − νdν ( x ) . We also have that the projection e from ℓ Γ ⊗ L ( X, ν ) → ℓ Γ is given by id ⊗ R . Thus, itfollows that the measure dϕ ( eE ( λ ) e ) agrees with dα ∗ ( µ × ν ), where α : Γ × X → R > is theRadon-Nikodym cocycle, α ( γ, x ) = dγ − νdν ( x ).In this case we then have h ϕ ( L Γ ⊂ L ∞ ( X, ν ) ⋊ Γ , ζ ) = − Z log( λ ) dϕ ( eE ( λ ) e )= − Z Z log (cid:18) dγ − νdν ( x ) (cid:19) d ( ν × µ ) = h µ ( a, ν ) . Lemma 5.4.
Let ϕ ∈ S τ ( B ( L ( M, τ ))) be a normal hyperstate and write ϕ in a standardform ϕ ( T ) = P n h T b z ∗ n , b z ∗ n i . Suppose A is a C ∗ -algebra with M ⊂ A and ζ ∈ S τ ( A ) is ahyperstate. Then if h ϕ ( M ⊂ A , ζ ) < ∞ we have that z ∗ n ζ ∈ D (log ∆ ζ ) for each n and h ϕ ( M ⊂ A , ζ ) = X n h log ∆ ζ z ∗ n ζ , z ∗ n ζ i = i lim t → t X n ( ζ ( z n σ ζt ( z ∗ n )) − . Proof. As A ζ forms a core for S ζ we get that z ∗ n ζ ∈ D (log(∆ ζ )). Also, we know thatlim t → ∆ itζ − t ξ = i log(∆ ζ ) ξ , for all ξ ∈ D (∆ ζ ). So, we have that h ϕ ( M ⊂ A , ζ ) = − ϕ ( e log(∆ ζ ) e ) = X n h log ∆ ζ z ∗ n ζ , z ∗ n ζ i = i X n h z n lim t → ∆ itζ − t z ∗ n ζ , ζ i = i lim t → t X n ( ζ ( z n σ ζt ( z ∗ n )) − . (cid:3) Example 5.5.
Fix two normal hyperstates ϕ, ζ ∈ S τ ( B ( L ( M, τ ))) such that ϕ is regu-lar, and ζ is faithful, and consider the case A = B ( L ( M, τ )). Then the density opera-tor A ζ is injective with dense range and the modular operator on L ( B ( L ( M, τ )) , ζ ) isgiven by ∆ ζ ( T ζ ) = A ζ T A − ζ ζ , for T ∈ B ( L ( M, τ )) such that T ζ ∈ D (∆ ζ ), so thatlog(∆ ζ )( T ζ ) = (Ad(log A ζ ) T )1 ζ , where Ad(log A ζ ) T = (log A ζ ) T − T (log A ζ ).We also have that the projection e : L ( B ( L ( M, τ )) , ζ ) → L ( M, τ ) is given by e ( T ζ ) = P ζ ( T )ˆ1. Therefore, e log ∆ ζ ex ˆ1 = P ζ (Ad(log A ζ ) x )ˆ1 = P ζ (Ad(log A ζ )) x ˆ1. Hence, h ϕ ( M ⊂ B ( L ( M, τ )) , ζ ) = ϕ ( P ζ (Ad(log A ζ )))= Tr( A ϕ ∗ ζ Ad(log A ζ ))= Tr( A ϕ ∗ ζ log A ζ ) − h log A ζ ˆ1 , ˆ1 i . Where the last equality follows since ϕ is regular.We recall the following two lemmas from works of D.Petz [Pet86]. Lemma 5.6.
Let ∆ j be positive, self adjoint operators on H j , j = 1 , . If T : H → H isa bounded operator such that: • T ( D (∆ )) ⊆ D (∆ ) • || ∆ T ξ || ≤ || T || · || ∆ ξ || ( ξ ∈ D (∆ ) ),then we have for each t ∈ [0 , , and ξ ∈ D (∆ t ) , || ∆ t T ξ || ≤ || T || · || ∆ t ξ || Lemma 5.7.
Let ∆ be a positive self adjoint operator and ξ ∈ D (∆) . Then: lim t → || ∆ t/ ξ || − || ξ || t exists. It’s finite or −∞ and equals ∞ R log λd h E λ ξ, ξ i where ∞ R log λdE λ is the spectralresolution of ∆. OISSON BOUNDARIES OF II FACTORS 19
Corollary 5.8. h ϕ ( M ⊂ A , ζ ) = − lim t → ∞ P k =1 || ∆ t/ ϕ ez ∗ n ˆ1 || − || ez ∗ n ˆ1 || t Lemma 5.9. h ϕ ( M ⊂ A , ζ ) ≥ Proof.
Let P ζ ( T ) = eT e for T ∈ A . h ϕ ( M ⊂ A , ζ ) = lim n →∞ ϕ ( − e log ∆ n e ) = − lim n →∞ hP ϕ ◦ P ζ (log ∆ n )ˆ1 , ˆ1 i ≥ lim n →∞ −h log( P ϕ ◦P ζ (∆ n ))ˆ1 , ˆ1 i (using the operator Jensen’s inequality; recall that log is operator concave).Now, e ∆ n e ≤ e ∆ e = I . So, P ϕ ◦ P ζ (∆ n ) ≤ I . As log is operator monotone, we get thatlog( P ϕ ◦ P ζ (∆ n )) ≤ log( I ) = 0. Hence we are done. (cid:3) Theorem 5.10.
Let ϕ, ψ ∈ S τ ( B ( L ( M, τ ))) be two normal hyperstates such that ψ isregular, and suppose A is a C ∗ -algebra with M ⊂ A , and ζ ∈ S τ ( A ) is a faithful hyperstatewhich is ψ -stationary. Then h ϕ ∗ ψ ( M ⊂ A , ζ ) = h ϕ ( M ⊂ A , ζ ) + h ψ ( M ⊂ A , ζ ) . Proof.
Let P ϕ and P ψ be the corresponding u.c.p. maps. Let P ϕ ( T ) = P k µ k J a ∗ k J T J a k J and P ψ ( T ) = P l ν l J b ∗ l J T J b l J . We shall denote the projection from L ( A , ζ ) to L ( M, τ )by e and ∆ ζ by ∆. We then have: h ϕ ( M ⊂ A , ζ ) = i lim t → ϕ ( e ∆ it e − t ) = i lim t → t ϕ ( e ∆ it e − i lim t → t ( X k µ k h (∆ it − a ∗ k ζ , a ∗ k ζ i )Similarly, h ψ ( M ⊂ A , ζ ) = i lim t → t ( X l ν l h (∆ it − b ∗ l ζ , b ∗ l ζ i )and, h ϕ ∗ ψ ( M ⊂ A , ζ ) = i lim t → t ( X k,l µ k ν l h (∆ it − a ∗ k b ∗ l ζ , a ∗ k b ∗ l ζ i )= i lim t → t ( X k,l µ k ν l h ( b l a k σ t ( a ∗ k b ∗ l )1 ζ , ζ i − t → t ( P k,l µ k ν l h b l a k σ t ( a ∗ k b ∗ l )1 ζ , ζ i − P k,l µ k ν l h b l σ t ( b ∗ l ) σ t ( a ∗ k )1 ζ , ζ i ) =0 . Let y t = a k σ t ( a ∗ k ). Note that y t → a k a ∗ k as t →
0, in SOT. We have: y t σ t ( b ∗ l ) − σ t ( b ∗ l ) y t = y t σ t ( b ∗ l ) − y t b ∗ l + y t b ∗ l − σ t ( b ∗ l ) y t = y t ( σ t ( b ∗ l ) − b ∗ l ) + ( y t b ∗ l − b ∗ l y t ) + ( b ∗ l − σ t ( b ∗ l )) y t Now,1 t ( X k,l µ k ν l h ( y t b ∗ l − b ∗ l y t )1 ζ , b ∗ l ζ i = 1 t ( X k,l µ k ν l h b l y t b ∗ l ζ , ζ i − t ( X k,l µ k ν l h y t ζ , b l b ∗ l ζ i = 1 t X k µ k h ( X l ν l b l y t b ∗ l )1 ζ , ζ i − t X k µ k h y t ζ , ζ i = 1 t h y t ζ , ζ i − t h y t ζ , ζ i = 0 , where the second to last equality holds by ψ -stationarity of ζ .Also, lim t → t ( y t ( σ t ( b ∗ l ) − b ∗ l )) exists, and hencelim t → t ( X k,l µ k ν l h b l a k σ t ( a ∗ k b ∗ l )1 ζ , ζ i − X k,l µ k ν l h b l σ t ( b ∗ l ) σ t ( a ∗ k )1 ζ , ζ i ) = 0 . So, we get that h ϕ ∗ ψ ( M ⊂ A , ζ ) = i lim t → t ( X k,l µ k ν l h ( b l σ t ( b ∗ l ) a k σ t ( a ∗ k ) − ζ , ζ i = i lim t → t ( X k,l µ k ν l [ h ( b l σ t ( b ∗ l ) − ζ , ζ i + h ( a k σ t ( a ∗ k ) − ζ , ζ i + h ( a k σ t ( a ∗ k ) − ζ , ( b l σ t ( b ∗ l ) − ∗ ζ i ]The first term equals h ϕ ( M ⊂ A , ζ ), while second term equals h ψ ( M ⊂ A , ζ ), and the thirdterm equals zero, as lim t → t ( a k σ t ( a ∗ k ) − ζ exists, while lim t → P l ν l ( b l σ t ( b ∗ l ) − ∗ ζ = 0 . (cid:3) Corollary 5.11.
Let ϕ ∈ S τ ( B ( L ( M, τ ))) be a regular normal hyperstate and suppose A is a C ∗ -algebra with M ⊂ A , and ζ ∈ S τ ( A ) is a faithful ϕ -stationary hyperstate, then for n ≥ we have h ϕ ∗ n ( M ⊂ A , ζ ) = nh ϕ ( M ⊂ A , ζ ) . Lemma 5.12. h ϕ ( M ⊂ A , ζ ) ≤ H ( ϕ ) Proof.
Let P ϕ ( T ) = P k µ k b k T b ∗ k . Let a k = J b k J ∈ M . It follows from 5.7 that H ( ϕ ) = − lim t → ∞ P k =1 µ k || A t/ ϕ a ∗ k ˆ1 || − || a ∗ k ˆ1 || t . So by corollary 5.8 it’s enough to show thatlim t → ∞ P k =1 µ k || A t/ ϕ a ∗ k ˆ1 || − || a ∗ k ˆ1 || t ≤ lim t → ∞ P k =1 µ k || ∆ t/ ϕ ea ∗ k ˆ1 || − || ea ∗ k ˆ1 || t . So, it’s enough to show that || A t/ ϕ a k ˆ1 || ≤ || ∆ t/ ζ a k ζ || FACTORS 21
Define T : L ( A , ζ ) → L ( M, τ ) by T ( a ζ ) = P ζ ( a )ˆ1. Then || T || = 1, as || T (1 ζ ) || = 1 and ||P ζ || ≤ T takes D (∆ ζ ) into D ( A ϕ ) = L ( M, τ ). By lemma 5.6 it’s enough to show: || A / ϕ T ξ || ≤ || ∆ / ξ || for all ξ ∈ D (∆) . In fact it’s enough to show the above for all vectors in a core for D (∆). Recall that A ζ forms a core for D (∆). So, we only need to show || A / ϕ T a ζ || ≤ || ∆ / a ζ || Now we have: || ∆ / a ζ || = h ∆ / a ζ , ∆ / a ζ i = h J Sa ζ , J Sa ζ i = h J a ∗ ζ , J a ∗ ζ i = h a ∗ ζ , a ∗ ζ i = ζ ( aa ∗ )= hP ζ ( aa ∗ )ˆ1 , ˆ1 i We also have P ϕ ◦ P ζ = P ζ = ⇒ ϕ ◦ P ζ = ζ . Now: || A / ϕ T a ζ || = h A / ϕ P ζ ( a )ˆ1 , A / ϕ P ζ ( a )ˆ1 i = h A ϕ P ζ ( a )ˆ1 , P ζ ( a )ˆ1 i = hP ζ ( a ) ∗ A ϕ P ζ ( a )ˆ1 , ˆ1 i ≤ T r ( P ζ ( a ) ∗ A ϕ P ζ )= T r ( A ϕ P ζ ( a ) P ζ ( a ∗ )) ≤ T r ( A ϕ P ζ ( aa ∗ )= h Φ ◦ P ζ ( aa ∗ )ˆ1 , ˆ1 i = hP ζ ( aa ∗ )ˆ1 , ˆ1 i = ζ ( aa ∗ ) = || ∆ / a ζ || . Hence we are done. (cid:3)
Corollary 5.13. h ϕ ( M ⊂ A , ζ ) ≤ h ( ϕ ) Proof.
By lemma 5.12, we have that h ϕ ∗ n ( M ⊂ A , ζ ) ≤ H ( ϕ ∗ n ). By corollary 5.11 we havethat h ϕ ∗ n ( M ⊂ A , ζ ) = nh ϕ ( M ⊂ A , ζ ). So we get, h ϕ ( M ⊂ A , ζ ) ≤ H ( ϕ ∗ n ) n → h ( ϕ ) . (cid:3) Lemma 5.14. h ϕ ( M ⊂ A , ζ ) = 0 if and only if there exists a normal ζ preserving condi-tional expectation from A to M .Proof. Let E : A → M be a normal ζ preserving conditional expectation. Then, we knowthat σ ζt ( m ) = m for all m ∈ M . Hence, h ϕ ( M ⊂ A , ζ ) = i lim t → t X k h (∆ it − a ∗ k ζ , a ∗ k ζ i = i lim t → t X k h σ t ( a ∗ k )1 ζ , a ∗ k ζ i − . Conversely, suppose h ϕ ( M ⊂ A , ζ ) = 0. Let ∆ ζ = ∆ and let ∆ = R ∞ λdλ be it’s spectralresolution. Let ∆ n = R n /n λdλ , n ≥ n converges to ∆in the resolvent sense. As usual, we denote by e the projection from L ( A , ζ ) to L ( M, τ ).We have that I = e ∆ e ≥ e ∆ n e for all n . So, ( I + t ) − ≤ ( e ∆ n e + t ) − ≤ e (∆ n + t ) − e for all n and for all t >
0. Taking limits as n → ∞ , we get ( I + t ) − ≤ e (∆ + t ) − e . Now weshall use the following integral representation of log:log( x ) = Z ∞ [(1 + t ) − − ( x + t ) − ] dt So that h ϕ ( M ⊂ A , ζ ) = − Z ∞ X k h e [( I + t ) − − (∆ + t ) − ] ea ∗ k ˆ1 , a ∗ k ˆ1 i . From h ϕ ( M ⊂ A , ζ ) = 0 and the above discussion, we deduce that:( I + t ) − a ∗ k ζ = (∆ + t ) − a ∗ k ζ for almost all t >
0, and hence by continuity, for all t >
0. This implies that ∆ it a ∗ k ζ = a ∗ k ζ ,which implies that σ ζt ( a ∗ k ) = a ∗ k and hence σ ζt ( m ) = m for all m ∈ M , as ϕ is generating.This implies the existence of a ζ preserving conditional expectation from A to M . (cid:3) Corollary 5.15.
Har( B ( L M, τ ) , P ϕ ) = M if and only if h ϕ ( M ⊂ B ϕ , ζ ) = 0 , where B ϕ denotes the Poisson boundary with respect to ϕ .Proof. If h ϕ ( M ⊂ B , ζ ) = 0, then by lemma 5.14 there exists a conditional expectation E : B → M , given by E ( b ) = ebe = P ( b ). So,Har( P ϕ ) = P ( B ϕ ) = M. Conversely, if Har( B ( L M, τ ) , P ϕ ) = M then ∆ ζ = I and hence h ϕ ( M ⊂ B ϕ , ζ ) = 0 (cid:3) Corollary 5.16.
Har( P ϕ ) = M if h ( ϕ ) = 0 .Proof. Since 0 ≤ h ϕ ( M ⊂ B ϕ , ζ ) ≤ h ( ϕ ), this result follows from Corollary 5.15. (cid:3) An entropy gap for property (T) factors
If (
M, τ ) is a tracial von Neumann algebra, then a Hilbert M -bimodule consists ofa Hilbert space H , together with commuting normal representations L : M → B ( H ), R : M op → B ( H ). We will sometimes simplify notation by writing xξy for the vec-tor L ( x ) R ( y op ) ξ . A vector ξ ∈ H is left (resp. right) tracial if h xξ, ξ i = τ ( x ) (resp. h ξx, ξ i = τ ( x )) for all x ∈ M . A vector is bi-tracial if it is both left and right tracial.A vector ξ ∈ H is central if xξ = ξx for all x ∈ M . Note that if ξ is a unit central vectorthen x
7→ h xξ, ξ i gives a normal trace on M .The von Neumann algebra M has property (T) if for any sequence of Hilbert bimodules H n , and ξ n ∈ H n bi-triacial vectors, such that k xξ n − ξ n x k → x ∈ M , then wehave k ξ n − P ( ξ n ) k →
0, where P is the projection onto the space of central vectors. Thisis independent of the normal faithful trace τ [Pop06, Proposition 4.1]. Property (T) wasfirst introduced in the factor case by Connes and Jones [CJ85] where they showed that foran ICC group Γ, the group von Neumann algebra L Γ has property (T) if and only if Γ hasKazhdan’s property (T) [Kaˇz67]. Their proof works equally well in the general case whenΓ is not necessarily ICC.We now suppose that M is finitely generated as a von Neumann algebra. Take { a k } nk ⊂ M a finite generating set such that P nk =1 a ∗ k a k = P nk =1 a k a ∗ k = 1, and let B ( L ( M, τ )) ∋ T OISSON BOUNDARIES OF II FACTORS 23 ϕ ( T ) = P nk =1 h T b a ∗ k , b a ∗ k i denote the associated normal regular hyperstate. For a fixed Hilbertbimodule H we define ∇ L , ∇ R : H → H ⊕ n by ∇ L ( ξ ) = ⊕ a k ξ ∇ R ( ξ ) = ⊕ ξa k . Note that we have k∇ L ( ξ ) k = n X k =1 k a k ξ k = * n X k =1 a ∗ k a k ξ, ξ + = k ξ k , and we similarly have k∇ R ( ξ ) k = * n X k =1 ξa k a ∗ k , ξ + = k ξ k . Thus ∇ L and ∇ R are both isometries. We let T denote the operator given by T ξ = P nk =1 a ∗ k ξa k . Note that T = ∇ ∗ R ∇ L and hence T is a contraction.Suppose now that M ⊂ A is an inclusion of von Neumann algebras and ζ ∈ A ∗ is a faithfulnormal hyperstate. We may then consider the Hilbert space L ( A , ζ ) which is naturally aHilbert M -bimodule where the left action is given by left multiplication L ( x )ˆ a = c xa , andthe right action is given by R ( x op ) = J L ( x ∗ ) J . In this case the vector ˆ1 is clearly lefttracial, and we also have J x ∗ J ˆ1 = ∆ / x ˆ1 from which it follows that ˆ1 is also right tracial.If ξ ∈ L ( A , ζ ) is a unit M -central vector, then τ ( x ) = h xξ , ξ i defines a normal trace on M . We let s ∈ Z ( M ) denote the support of τ . Lemma 6.1.
Let ( M, τ ) , ϕ , and ( A , ζ ) be as given above, then h ϕ ( M ⊂ A , ζ ) ≥ − h T ζ , ζ i . Proof.
First, note that L a ∗ k R a k ζ = a ∗ k ∆ / a k ζ . Now, − h T ζ , ζ i = − n X k =1 h a ∗ k ∆ / a k ˆ1 , ˆ1 i )= − n →∞ log( n X k =1 h a ∗ k ∆ / n a k ˆ1 , ˆ1 i ) ≤ lim n →∞ n X k =1 h a ∗ k log(∆ n ) a k ˆ1 , ˆ1 i = h ϕ ( M ⊂ A , ζ ) , where the inequality follows from Jensen’s operator inequality. (cid:3) Theorem 6.2.
Let M be a II factor generated as a von Neumann algebra by { a k } nk =1 suchthat P nk =1 a ∗ k a k = P nk =1 a k a ∗ k = 1 . Let B ( L ( M, τ )) ∋ T ϕ ( T ) = n X k =1 h T b a ∗ k , b a ∗ k i denote the associated normal regular hyperstate. If M has property (T), then there exists c > such that if M ⊂ A is any irreducible inclusion having no normal conditional expectationfrom A to M , and if ζ ∈ A ∗ any faithful normal hyperstate, then h ϕ ( M ⊂ A , ζ ) ≥ c . Proof.
Suppose M has property (T) and there is a sequence of irreducible inclusions M ⊂A m , and normal faithful hyperstates ζ m ∈ A m , such that h ϕ ( M ⊂ A m , ζ m ) →
0. Thenby Lemma 6.1 we have that h T ζ m , ζ m i →
1, and hence P nk =1 k a k ζ m − ζ m a k k = 2 − h T ζ m , ζ m i →
0. Since M has property (T) it then follows that for m large enough thereexists a unit M -central vector ξ ∈ L ( A m , ζ m ). If we let ˜ ζ denote the state on A m given by˜ ζ ( a ) = h aξ, ξ i , then as ξ is M -central we have that ˜ ζ gives an M -hypertrace on A m . Thus,there exists a corresponding normal conditional expectation form A m to M , for all m largeenough. (cid:3) Acknowledgments
SD is immensely grateful to Darren Creutz for explaining the theory of Poisson boundariesof groups to him, and for many useful remarks and stimulating conversations about earlierdrafts of this paper. SD would like to gratefully acknowledge many helpful conversationswith Vaughan Jones and Ionut Chifan regarding this paper. SD would also like to thankBen Hayes and Krishnendu Khan for various discussions in and around the contents of thispaper. JP would like to thank Sorin Popa for useful comments regarding this paper.7.
Appendix: Minimal dilations and boundaries of u.c.p. maps
We include in this appendix a proof of Izumi’s result from [Izu02] that for a von Neumannalgebra (or even an arbitrary C ∗ -algebra) A , and a u.c.p. map φ : A → A , the operator spaceHar( A, φ ) has a C ∗ -algebraic structure. We take the approach in [Izu12] where Har( A, φ )is shown to be completely isometric to the ∗ -algebra of fixed points associated to a ∗ -endomorphism which dialates the u.c.p. map. There are several proofs of the existence ofsuch a dilation, the first proof is by Bhat in [Bha99] in the setting of completely positivesemigroups, building on work from [Bha96], [BP94], and [BP95], and then later proofs weregiven in [BS00], [MS02], and Chapter 8 of [Arv03]. Our reason for including an additionalproof is that it is perhaps more elementary than previous proofs, being based on a simpleidea of iterating the Stinespring dilation [Sti55]. Lemma 7.1. 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( V BV ∗ , A ) V = ∗ - alg( B, V ∗ AV ) .Proof. Using the fact that V ∗ V = 1, this follows easily by induction on the length ofalternating products for monomials in V BV ∗ , and A . (cid:3) If A ⊂ B ( H ) is a C ∗ -algebra, and φ : A → A is a unital completely positive map,then one can iterate Stinespring’s dilation as follows: Lemma 7.2.
Suppose A ⊂ B ( H ) is a unital C ∗ -algebra, and φ : A → A is a unitalcompletely positive map. Then there exists a sequence whose entries consist of: ( ) a Hilbert space H n ; ( ) an isometry V n : H n − → H n ; ( ) a unital C ∗ -algebra A n ⊂ B ( H n ) ; ( ) a unital representation π n : A n − → B ( H n ) , such that π n ( A n − ) , and V n A n − V ∗ n gen-erate A n ; ( ) a unital completely positive map φ n : A n → A n ; OISSON BOUNDARIES OF II FACTORS 25 such that the following relationships are satisfied for each n ∈ N , x ∈ A n − : V ∗ n π n ( x ) V n = φ n − ( x );(5) V ∗ n A n V n = A n − ;(6) φ n ( π n ( x )) = π n ( φ n − ( x ));(7) π n +1 ( V n xV ∗ n ) = V n +1 π n ( x ) V ∗ n +1 . (8) Moreover, for each n ∈ N we have that the central support of V n V ∗ n in A ′′ n is . Also, if A is a von Neumann algebra and φ is normal then A n will also be a von Neumann algebraand π n and φ n will be normal for each n ∈ N .Proof. We will first construct the objects and show the relationships (5), (6), and (7) byinduction, with the base case being vacuous, and we will then show that (8) also holds forall n ∈ N . So suppose n ∈ N and that (5), (6), and (7) hold for all m < n , (we leave V undefined).From the proof of Stinespring’s Dilation Theorem we may construct a Hilbert space H n by separating and completing the vector space A n − ⊗H n − with respect to the non-negativedefinite sesquilinear form satisfying h a ⊗ ξ, b ⊗ η i = h φ n − ( b ∗ a ) ξ, η i , for all a, b ∈ A n − , ξ, η ∈ H n − .We also obtain a partial isometry V n : H n − → H n from the formula V n ( ξ ) = 1 ⊗ ξ, for ξ ∈ H n − .We obtain a representation π n : A n − → B ( H n ) (which is normal when A is a vonNeumann algebra and φ is normal) from the formula π n ( x )( a ⊗ ξ ) = ( xa ) ⊗ ξ, for x, a ∈ A n − , ξ ∈ H n − . And recall the fundamental relationship V ∗ n π n ( x ) V n = φ n − ( x )for all x ∈ A n − , which establishes (5).If we let A n be the C ∗ -algebra generated by π n ( A n − ) and V n A n − V ∗ n , then π n : A n − → A n , and from Lemma 7.1 we have that V ∗ n A n V n is generated by V ∗ n π n ( A n − ) V n and A n − .However, V ∗ n π n ( A n − ) V n = φ n − ( A n − ) ⊂ A n − , hence V ∗ n A n V n = A n − , establishing (6).Also, when A is a von Neumann algebra and π n is normal it then follows easily that A n isthen also a von Neumann algebra.Also note that π n ( A n − ) V n V ∗ n H n is dense in H n , and so since π n ( A n − ) ⊂ A n we havethat the central support of V n V ∗ n in A ′′ n is 1.We then define φ n : A n → A n by φ n ( x ) = π n ( V ∗ n xV n ), for x ∈ A n . This is well definedsince V ∗ n A n V n = A n − , unital, and completely positive. Note that for x ∈ A n − we have φ n ( π n ( x )) = π n ( V ∗ n π n ( x ) V n ) = π n ( φ n − ( x )), establishing (7). Having established (5), (6), and (7) for all n ∈ N , we now show that (8) holds as well.For this, notice first that for a, b ∈ A n , x ∈ A n − , and ξ, η ∈ H n we have h π n +1 ( V n xV ∗ n )( a ⊗ ξ ) , b ⊗ η i = h V n xV ∗ n a ⊗ ξ, b ⊗ η i = h φ n ( b ∗ V n xV ∗ n a ) ξ, η i = h π n ( V ∗ n b ∗ V n xV ∗ n aV n ) ξ, η i = h ⊗ π n ( xV ∗ n aV n ) ξ, b ⊗ η i . Setting x = 1 and using that V ∗ n +1 (1 ⊗ ζ ) = ζ for each ζ ∈ H n , we see that( V n +1 V ∗ n +1 ) π n +1 ( V n V ∗ n )( a ⊗ ξ ) = ( V n +1 V ∗ n +1 )(1 ⊗ π n ( V ∗ n aV n ) ξ )= 1 ⊗ π n ( V ∗ n aV n ) ξ = π n +1 ( V n V ∗ n )( a ⊗ ξ ) , and hence π n +1 ( V n V ∗ n ) ≤ V n +1 V ∗ n +1 . If instead we set a = 1 then we have V n +1 π n ( x ) ξ = 1 ⊗ π n ( x ) ξ = π n +1 ( V n xV ∗ n ) V n +1 ξ, and so V n +1 π n ( x ) = π n +1 ( V n xV ∗ n ) V n +1 . Multiplying on the right by V ∗ n +1 and using that π n ( V n V ∗ n ) ≤ V n +1 V ∗ n +1 then gives V n +1 π n ( x ) V ∗ n +1 = π n +1 ( V n xV ∗ n ). (cid:3) Theorem 7.3 (Bhat [Bha99]) . Let A ⊂ B ( H ) be a unital C ∗ -algebra, and φ : A → A a unital completely positive map. Then there exists ( ) a Hilbert space K ; ( ) an isometry W : H → K ; ( ) a C ∗ -algebra B ⊂ B ( K ) ; ( ) a unital ∗ -endomorphism α : B → B ;such that W ∗ BW = A , and for all x ∈ A we have φ k ( x ) = W ∗ α k ( W xW ∗ ) W. Moreover, we have that the central support of P in B ′′ is , and for y ∈ B ( K ) we have y ∈ B if and only if α k ( W W ∗ ) yα k ( W W ∗ ) ∈ α k ( W A W ∗ ) for all k ≥ . Also, if A is avon Neumann algebra and φ 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 thedirected limit of the Hilbert spaces H n with respect to the inclusions V n +1 : H n → H n +1 . Wedenote by W n : H n → K the associated sequence of isometries satisfying W ∗ n +1 W n = V n +1 ,for n ∈ N , and we set P n = W n W ∗ n , an increasing sequence of projections.From (6) we have that P n − W n A n W ∗ n P n − = W n − A n − W ∗ n − , and hence if we definethe C ∗ -algebra B = { x ∈ B ( K ) | W ∗ n xW n ∈ A n , n ≥ } , then we have W ∗ n BW n = A n , forall n ≥
0. Also, if A is a von Neumann algebra, then so is A n for each n ∈ N and fromthis it follows easily that B is also a von Neumann algebra.We define the unital ∗ -endomorphism α : B → B (which is normal when A is a vonNeumann algebra and φ is normal) by the formula α ( x ) = lim n →∞ W n +1 π n +1 ( W ∗ n xW n ) W n +1 , OISSON BOUNDARIES OF II FACTORS 27 where the limit is taken in the strong operator topology. Note that α ( P n ) = P n +1 ≥ P n .From (8) we see that in general, the strong operator topology limit exists in B , and thatfor x ∈ A n ∼ = P n A ∞ P n the limit stabilizes as α ( W n xW ∗ n ) = W n +1 π n +1 ( x ) W ∗ n +1 .From (5) we see that for n ≥
0, and x ∈ A n we have P n α ( W n xW ∗ n ) P n = W n W ∗ n W n +1 π n +1 ( x ) W ∗ n +1 W n W ∗ n = W n V ∗ n +1 π n +1 ( x ) V n +1 W ∗ n = W n φ n ( x ) W ∗ n . By induction we then see that also for k >
1, and x ∈ A we have P α k ( W xW ∗ ) P = P α k − ( P α ( W xW ∗ ) P ) P = P α k − ( W φ ( x ) W ∗ ) P = W φ k ( x ) W ∗ . By the previous lemma we have that the central support of P n in W n A ′′ n W ∗ n is P n +1 .Hence it follows that the central support of P in B is 1. (cid:3) Poisson boundaries of u.c.p. maps. 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 becoinvariant, if { φ n ( p ) } defines an increasing sequence of projections which strongly convergeto 1 in B ( H ), and such that for y ∈ B ( H ) we have y ∈ A if and only if φ n ( p ) yφ n ( p ) ∈ A for all n ≥
0. Note that for n ≥ φ n ( p ) is in the multiplicative domain for φ , and is againcoinvariant. We define φ p : pAp → pAp to be the map φ p ( x ) = pφ ( x ) p , then φ p is normalunital completely positive. Moreover, we have that φ kp ( x ) = pφ k ( x ) p for all x ∈ pAp , whichcan be seen by induction from pφ k ( x ) p = pφ k − ( p ) φ k ( x ) φ k − ( p ) p = pφ k − ( φ p ( x )) p. Theorem 7.4 (Prunaru [Pru12]) . Let A ⊂ B ( H ) be a unital C ∗ -algebra, φ : A → A aunital completely positive map, and p ∈ A a coinvariant projection. Then the map P :Har( A, φ ) → Har( pAp, φ p ) given by P ( x ) = pxp defines a completely positive isometricsurjection, between Har(
A, φ ) and Har( pAp, φ p ) .Moreover, if A is a von Neumann algebra and φ is normal then P is also normal.Proof. First note that P is well-defined since if x ∈ Har(
A, φ ) we have φ p ( pxp ) = pφ ( p ) xφ ( p ) p = pxp. Clearly P is completely positive (and normal in the case when A is a von Neumann algebraand φ is normal).To see that it is surjective, if x ∈ Har( pAp, φ p ) then consider the sequence φ n ( x ). Foreach m, n ≥
0, we have φ m ( p ) φ m + n ( x ) φ m ( p ) = φ m ( pφ n ( x ) p ) = φ m ( φ np ( x )) = φ m ( x ) . It follows that { φ n ( x ) } converges in the strong operator topology to an element y ∈ B ( H )such that φ m ( p ) yφ m ( p ) = φ m ( 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 limitlim n →∞ φ ( φ n ( p ) zφ n ( p )) = φ n +1 ( p ) φ ( z ) φ n +1 ( p ) = φ ( z ) , and hence φ ( y ) = lim m →∞ φ ( φ m ( p ) yφ m ( p )) = lim m →∞ φ m +1 ( x ) = y. Thus P is surjective, and since φ n ( p ) converges strongly to 1, and each φ n ( p ) is in themultiplicative domain of φ , it follows that if x ∈ Har(
A, φ ) then φ n ( pxp ) converges stronglyto x and hence k x k = lim n →∞ k φ n ( pxp ) k ≤ k pxp k ≤ k x k . Thus, P is also isometric. (cid:3) Corollary 7.5 (Izumi [Izu02]) . Let A be a unital C ∗ -algebra, and φ : A → A a unital com-pletely positive map. Then there exists a C ∗ -algebra B and a completely positive isometricsurjection P : B → Har(
A, φ ) .Moreover B and P are unique in the sense that if ˜ B is another C ∗ -algebra, and P : ˜ B → Har(
A, φ ) is a completely positive isometric surjection, then P − ◦ P is an isomorphism.Also, if A is a von Neumann algebra and φ is normal, then B is also a von Neumannalgebra and P is normal.Proof. Note that we may assume A ⊂ B ( H ). Existence then follows by applying the previoustheorem to Bhat’s dilation. Uniqueness follows from [Cho74] (cid:3) Corollary 7.6 (Choi-Effros [CE77]) . Let A be a unital C ∗ -algebra and F ⊂ A an operatorsystem. 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 Neumannalgebra and F is weakly closed then this gives a von Neumann algebraic structure on F .Proof. When A is a C ∗ -algebra this follows from Corollary 7.5 since Har( A, E ) = F . Alsonote that since E n = E it follows from the proof of Theorem 7.4 that the product structurecoming 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 bySakai’s theorem. (cid:3) Proposition 7.7.
Let A be an abelian C ∗ -algebra and φ : A → A a normal unital completelypositive map. Then the Poisson boundary of φ is also abelian.Proof. Let B be the Poisson boundary of φ , and let P : B → Har(
A, φ ) be the Poissontransform. If C is a C ∗ -algebra and ψ : C → B is a positive map then P ◦ ψ : C → Har(
A, φ ) ⊂ A is positive, and since A is abelian it is then completely positive Hence, ψ is also completely positive. Since every positive map from a C ∗ -algebra to B is completelypositive it then follows that B is abelian. (cid:3) Example 7.8.
Let Γ be a discrete group and µ ∈ Prob(Γ) a probability measure on Γsuch that the support of µ generates Γ. Then on ℓ ∞ Γ we may consider the normal unital(completely) positive map φ µ given by φ µ ( f ) = µ ∗ f , where µ ∗ f is the convolution( µ ∗ f )( x ) = R f ( g − x ) dµ ( g ). Then Har( µ ) = Har( ℓ ∞ Γ , φ µ ) has a unique von Neumann OISSON BOUNDARIES OF II FACTORS 29 algebraic structure which is abelian by the previous proposition. Notice that Γ acts onHar( µ ) by right translation, and since this action preserves positivity it follows from [Cho74]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. map ˜ ϕ : ℓ ∞ Γ ⋊ Γ → ℓ ∞ Γ ⋊ Γsuch that ˜ ϕ L Γ = id. Note that ℓ ∞ Γ ⋊ Γ ∼ = B ( ℓ Γ). It is an easy exercise to see that thePoisson boundary of ˜ φ is nothing but the crossed product Har( µ ) ⋊ Γ. References [Arv03] William Arveson,
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