This relates naturally to the study of von Neumann algebras, which we shall describe. A von Neumann algebra M is called a factor if the only operators commute with all Mare scalar multiples of identity. In general, if a von Neumann algebra admits a faithful trace stateτ, we say that the Mis affinitevon Neumann algebra.
As it turns out, every abelian von Neumann algebra (on a separable Hilbert space) is isomorphic to L∞(X,µ) for some space(X,µ). Therefore, the study of von Neumann algebras can be considered a non-commutative gauge theory. Other exciting examples of von Neumann algebras are constructed by group representations and actions, as described below.
The first examples of von Neumann algebras other than the above examples were the so-called von Neumann group algebras of discrete groups. Thus, many questions in ergodic theory and group representations can naturally be studied in the context of von Neumann algebra.
Poisson boundaries of groups
The Poisson limit of a group, (B,β), provides a natural example of such a probability measure space on which the group acts. In general, the actions are not goal-preserving: indeed, in the case of non-receptive groups, the action on its Poisson limit can never be goal-preserving. The space of harmonic functions with this new product then becomes isomorphic to L∞(B,β), for some probability measure the space (B,β) - which we call the Poisson limit of the group G.
For the case of the free group of 2 generators, F2=ha,bi, with the uniform probability measure µ ona,a−1,b,b−1, the Poisson limit can be identified with the space of one-sided infinite reduced words, with a natural Borel- probability measure. This example suggests that the Poisson limit can be thought of as an "exit limit" for random walks on the Cayley graph of G. One of the most striking applications of Poisson limits was by Margulis in his proof of the Normal subgroup theorem ([ Mar78] [ Mar79]).
The tractability half of the result was based on looking at the action of Gon's Poisson limit (which is a tractable action). As we shall see, this was one of the key motivations for studying the Poisson limit in the non-commutative setting.
Poisson boundary of finite von Neumann algebras
A von Neumann algebra Mis is said to be admissible if it is injective as an operator system.
Hyperstates and bimodular u.c.p. maps
Poisson boundaries of II 1 factors
If ϕ ∈Sτ(B(L2(M,τ)) is a hyperstate, we define the Poisson limit Bϕ of M with respect to ϕ to be the non-commutative Poisson limit of the u.c.p. The Poisson limit contains M as a subalgebra and the inclusion (M ⊂ Bϕ) is determined to isomorphism by the property that there exists a completely positive isometric isomorphism P : Bϕ →Har(B(L2(M,τ)),Pϕ) which is restricted to the identity map on M. As Bϕ is isomorphic to Har(B(L2(M,τ)),Pϕ) as an operator system it follows that Bϕ is injective.
We say that ϕ is regular if the restriction of Pϕ to M0 preserves the canonical track on M0, and we say that ϕ generates if M is the largest∗-subalgebra of B(L2(M,τ)) found in Har(B( L2(M,τ)) is. ),Pϕ). Ifϕ is regular, then Pϕ∗(T) =R(JxJ)T(Jx∗J)dµ(x) and ϕ is symmetric if J∗µ =µ, where J is the adjoint operation. If the support of µ generates a weakly closed subalgebraM0 containing the identity such that M0 6=M, then [Jx∗J,eM0] =0 for every x in the support of µ.
Thus, if the support of µ generates a weakly dense subalgebra containing the identity, then we have that T ∈JMJ0=M, which shows that ϕ generates. Thus, Har(Pϕ,JMJ) is a von Neumann subalgebra of JMJby Lemma 3.1.5, which must be Z(M) since ϕ generates. Ifϕ is a normal hyperstate in Sτ(B(L2(M,τ))), then Pϕ :B(L2(M,τ))→B(L2(M,τ)) is a normal map, and so the dual mapPϕ∗ preserves the predual of B(L2(M,τ)) which we identify with the space of trace class operators.
If Pˆ1 denotes the orthogonal projection of rank one onto Cˆ1, then ϕ(T) =hPϕ(T)ˆ1,ˆ1i=Tr(Pϕ(T)Pˆ1), and therefore we see that Aϕ =Pϕ∗(Pˆ1). Suppose ϕ ∈Sτ(B(L2(M,τ))) is a normal hyperstate, then there exists a τ-orthogonal family{zn}nwhich gives a partition of the identity as1=∑nz∗nznso that. Since the set of operators of the form xPˆ1y is compact in the space of trace class operators, it follows that Pϕ(T) =∑nan(JynJ)T(Jy∗nJ)for allT ∈B(L2(M,τ)).
Since {zn}n was constructed above using any orthonormal basis of eigenvectors from Aϕ, the rest of the proposition follows easily. Remark: It follows from 3.1.4 that ϕ generates if and only if the weakly closed subalgebra generated by {zn}isM. This shows that ϕ is faithful to Har(Pϕ), and thus the stationary stateζ of Bϕ is faithful.
Construction of the boundary
This shows that ϕ is faithful on Har(Pϕ), and so the steady stateζ on Bϕ is faithful. andφ is normal). It follows that in the strong operator topology {φn(x)} converges to an element y∈B(H ) such that φm(p)yφm(p) =φm(x) for every m≥0, and so we have y∈A . To see that y∈Har(A,φ) we use that for allz∈Awe have the strong operator topology limit.
θ is therefore surjective, and since φn(p) strongly converges to 1, and every φn(p) is in the multiplicative domain of φ, it follows that if x∈Har(A,φ) then φn(pxp) strongly converges to x and therefore. 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 ˜ 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.
We refer to the C∗ algebraB from the previous corollary as the Poisson limit of φ and we refer to the mapθ as the Poisson transform. If E :A→F is a complete 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, this gives a von Neumann algebraic structure on F.
Note also that since En=E it follows from the proof of Theorem 3.1.9 that the product structure coming from the Poisson frontier is given by x·y=E(xy). If A is a von Neumann algebra and F is weakly closed, F has a predualF⊥={ϕ ∈A∗| ϕ(x) =0, for allx∈F} and therefore A is isomorphic to an algebraic von Neumann by Sakai's theorem. Note that if Ais aC∗-algebra,F ⊂On operator system, andE :A→F is fully positive withE|F =id, we still have a form of bimodularity forE if we provide F with the Choi-Effros product from Corollary 3.1. 11.
In this case even though bimodularity is about two different product structures, i.e., we have.
Bi-harmonic operators
We identify the Poisson limitBϕ with Har(B(L2(M,τ))), and let ζ denote the stationary state of Bϕ which is credible by Proposition 3.1.8. Since M0 =JMJ is isomorphic to L(S∞), by [KV83], there exists a regular, symmetric, generating hyperstateϕ such that Har(Pϕ) =M0.
Entropy
Asymptotic entropy
A Furstenberg type entropy
If Γ is a discrete group, μ ∈Prob(Γ) andΓya(X,ν) is an almost invariant action, then we can consider the state ϕ in B(`2Γ) given by ϕ(T) =RhTδγ,δγidµ (γ) , and we can consider the conditionζ in L∞(X,ν)oΓ⊂B(`2Γ⊗L2(X,ν)) given byζ ∑γ∈Γaγuγ. The modular operator ∆ζ is then related to the von Neumann algebra`∞Γ⊗L∞(X,ν), and we can calculate this directly as. Arrange two normal hyperstatesϕ,ζ∈Sτ(B(L2(M,τ))) such that ϕ is regular, andζ is faithful, and consider the case A =B(L2(M,τ)).
Then the density operator Aζ is injective with dense range and the modular operator on L2(B(L2(M,τ)),ζ) is given by ∆ζ(T1ζ) =AζTA−1. Let ϕ,ψ ∈Sτ(B(L2(M,τ))) be two normal hyperstates such that ψ is regular, and suppose that A is a C∗-algebra with M⊂A and ζ ∈Sτ(A) is faithful hyperstate, then . The first term is equal to shϕ(M⊂A,ζ), while the second term is equal to shψ(M⊂A,ζ) and the third term is equal to zero, as lim.
Letϕ ∈Sτ(B(L2(M,τ))) be a regular normal hyperstate and suppose A is a C∗-algebra with M⊂A, andζ ∈Sτ(A) is a faithful hyperstateϕ-stationary hyperstate, then for n≥ 1 we have.
An entropy gap for property (T) factors
Property (T) was first introduced in the factorial case by Connes and Jones [CJ85] where they showed that for an ICC group Γ, the von Neumann algebra group LΓ has property (T) if and only if Γ has the Kazhdan property ( T) [Kaˇz67]. A von Neumann tracial algebra M together with a hyperstateϕ is said to have an entropy gap if there exists a constant ε =ε(M,ϕ) such that for any stationary space (A,ζ) the Furstenberg entropy is at least ε. We will show that if M has property (T) in the sense of Connes and Jones, then (M,ϕ) has an entropy gap for every regular, symmetric hyperstate.
Since 1ζn are bi-tracial vectors, ||ak1ζn−1ζnak|| satisfies →0 (by convexity of Hilbert space) we get that there exists a central vector ψ (since generators M and M have property (T). The rest follows easily.
Rigidity for u.c.p. maps on the boundary
We clearly have Har(Pϕ1)⊗Har(Pϕ2)⊂Har(Pϕ1⊗Pϕ2), so we only need to show the inverse inclusion. DP17] Sayan Das and Jesse Peterson, Poisson limits of finite von Neumann algebras, in preparation, 2017. Izu99] Masaki Izumi, Actions of compact quantum groups on operator algebras, XIIth International Congress of Mathematical Physics (ICMP '97) (Brisbane ), Int.
Kaimanovich, Dual ergodicity of the poisson limit and applications to constrained cohomology, Geometric & Functional Analysis GAFA13(2003), no. Kaˇzdan, On the connection of the dual space of a group with the structure of its closed subgroups, Functional. Kra91] Jon Kraus, The Slice Mapping Problem and Approximation Properties, Journal of Functional Analysis no.
