Free Entropy Dimension in finite von Neumann Algebras. Junhao Shen

Transcription

1 Free Entropy Dimension in finite von Neumann Algebras Junhao Shen

2 von Neumann algebras Let H be a separable complex Hilbert space. Let B(H) be the set of all bounded linear operators from H to H. The adjoint of a bounded linear operator T is the operator T characterized by the identity < T v, w >=< v, T w > v, w H.

3 Weak operator topology (WOT) on B(H) is the topology such that a sequence (or a net) {T α } converges to T in the weak operator topology if and only if for all v 1, v 2 H. T α v 1, v 2 T v 1, v 2 A von Neumann algebra M is defined to be a selfadjoint subalgebra of B(H) which is closed in weak operator topology, i.e. (i) T M, then T M; and (ii) M W OT = M. Example: L (T, µ), where µ is the Lebesgue measure on the unit circle T.

4 Suppose M is a von Neumann algebra. Define M = {T B(H) ST = T S, S M} to be the commutant of M and Z(M) = M M to be the center of M. Factors are the von Neumann algebras whose centers are scalar multiples of the identity, i.e. Z(M) = CI. The factors are classified by means of a relative dimension function into type I n, II 1, II, III factors.

5 1. A factor M is a type I n factor if it is * isomorphic to B(H) for some n dimensional Hilbert space H. 2. A factor M is a type II 1 factor if and only if that it is infinite dimensional algebra and there is a positive linear mapping τ from M to C such that τ(ab) = τ(ba) for each A and B in M and τ(i) = 1. Such positive linear mapping τ is also called a trace on M. 3. A factor M is a type II factor if it is * isomorphic to R B(H) for some II 1 factor R and a infinite dimensional Hilbert space H. 4. All other factors are type III factors.

6 Examples of factors of type II 1 We assume that G is discrete countable group and the Hilbert space H is l 2 (G). For each g in G, let L g denote the left translation of functions in l 2 (G) by g 1. Then g L g is a faithful unitary representation of G on H. Let L(G) = alg{l g : g G} W OT be the von Neumann algebra generated by {L g : g G}. Proposition 1 (Murray and von Neumann) If G is an infinity conjugacy class group ( e g G, the cardinality of the set {h 1 gh h G} is infinite), then L(G) is a II 1 factor.

7 1. L(F (n)) (n 2), where F (n) is the free group with n generators. 2. L(Π), where Π is the permutation group of Z (consisting of those permutations that leave fixed all but a finite set of Z). 3. L(SL(3, Z)), where SL(3, Z) is the special linear group with integer entries.

8 2. Free entropy theory A result proved by Voiculescu using random matrices Proposition 2 Let L(F n ) be the free group factor on n generators with the tracial state τ, and u 1,..., u n be the standard generators of L(F n ). For each m, k 1 and ɛ > 0, let Ω m,ɛ (k) ={(U 1,..., U n ) U(k) n Then τ k (U η 1 i 1 U η p i p ) τ(u η 1 i 1 u η p i p ) < ɛ for all 1 p m, 1 i 1,..., i p n, {η 1,..., η p } {1, }}. where µ k U(k) n. lim µ k(ω m,ɛ (k)) = 1, k is normalized Haar measure on

9 Free entropy Let M k (C) be the k k full matrix algebra with entries in C and τ k be the normalized trace on M k (C), i.e., τ k = 1 ktr, where Tr is the usual trace on M k (C). Let Mk sa denote the self-adjoint complex matrices. The euclidean norm e on (Mk sa)n is given by (A 1,..., A n ) 2 e = Tr (A A2 n), for each (A 1,..., A n ) in (M sa k )n. Let Λ denote the Lebesgue measure on (M sa k )n induced by the euclidean norm e.

10 Voiculescu s microstate space Let (M, τ) be a free probability space. (In our case, M is a von Neumann algebra with a faithful normal tracial state τ). Let X 1,..., X n be self-adjoint elements in M. For ɛ, R > 0, m, k N, let Γ R (X 1,..., X n ; m, k, ɛ) be a subset of (M sa k )n consisting of all in (M sa k )n such that (A 1,..., A n ) τ(x i1... X ip ) τ k (A i1... A ip ) ɛ, for all 1 p m, 1 i 1,..., i p n and A j < R, 1 j m.

11 Then, we define successively, χ R (X 1,..., X n ; m, k, ɛ) = log Λ(Γ R (X 1,..., X n ; m, k, ɛ)), χ R (X 1,..., X n ; m, ɛ) = lim sup (k 2 χ R (X 1,..., X n ; m, k, ɛ) + n2 ) log k, k χ R (X 1,..., X n ) = inf {χ r (X 1,..., X n ; m, ɛ) : m N, ɛ > 0}, χ(x 1,..., X n ) = sup χ R (X 1,..., X n ). R>0 Proposition 3 Let R > max{ X 1,..., X n } be a positive number. Then χ(x 1,..., x n ) = lim sup ω 0 χ R(x 1,,..., x n ).

12 D. Voiculescu proved the following Theorem 1 Basic Properties of χ(x 1,..., X n ), 1. Upper Bound χ(x 1,..., X n ) 2 1 n log(2πen 1 C 2 ) where C 2 = τ(x X2 n). In particular χ(x 1,..., X n ) is either finite or. Proof: Let C 1 > τ(x X2 n) and Γ = {(A 1,..., A n ) (M sa k )n (A 1,..., A n ) e kc 1 }. Then when m is large enough and ɛ is small enough, we have Γ R (X 1,..., X n ; k, m, ɛ) Γ.

13 But λ(γ) = πnk2 /2 ( kc 1 ) nk2 Γ(1 + nk 2 /2). Using the sterling s formula, we have Γ(1 + nk 2 /2) nk 2 π (nk2 /2) nk2 /2. Thus e nk2 /2 lim sup( log(λ(γ)) k k 2 + n log k) n log(2πen 1 C1 2 ) 2. One Variable Case χ(x) = log s t dµ(s)dµ(t) log 2π, where µ is the distribution of X. 3. If X 1,..., X n are freely independent, then, χ(x 1,..., X n ) = χ(x 1 ) + + χ(x n )

14 Free entropy dimension Covering number in complex matrix algebras Let M k (C) be the k k full matrix algebra with entries in C, and τ k be the normalized trace on M k (C), i.e., τ k = 1 kt r, where T r is the usual trace on M k (C). Let M k (C) n denote the direct sum of n copies of M k (C). Let 2 denote the trace norm induced by τ k on M k (C) n, i.e., (A 1,..., A n ) 2 2 = τ k(a 1 A 1)+...+τ k (A na n ) for all (A 1,..., A n ) in M k (C) n.

15 For every ω > 0, we define the ω- 2 -ball Ball(B 1,..., B n ; ω, 2 ) centered at (B 1,..., B n ) in M k (C) n to be the subset of M k (C) n consisting of all (A 1,..., A n ) in M k (C) n such that (A 1,..., A n ) (B 1,..., B n ) 2 < ω. Definition 1 Suppose that Σ is a set in M k (C) n. We define ν 2 (Σ, ω) to be the minimal number of ω- 2 -balls that consist a covering of Σ in M k (C) n.

16 Voiculescu s microstate spaces Let (M, τ) be a finite von Neumann algebra, X 1,..., X n be elements in M. For ɛ, R > 0, m, k N, let Γ R (X 1,..., X n ; m, k, ɛ) be a subset of (M sa k )n consisting of all in (M sa k )n such that (A 1,..., A n ) τ(x i1... X in ) τ k (A η 1 i 1... A η n in ) ɛ, for all 1 p m, (i 1,..., i p ) {1,..., n} p, η 1,..., η n {1, } and A j < R, 1 j m.

17 Now we define, successively, δ 0 (x 1,..., x n ; ω, R, m, ɛ) log(ν = lim sup 2 (Γ R (x 1,..., x n ; m, k, ɛ), ω)) k k 2 log ω δ 0 (x 1,..., x n ; ω, R) = inf m N,ɛ>0 δ 0(x 1,..., x n ; ω, R, m, ɛ) δ 0 (x 1,..., x n ; ω) = sup δ 0 (x 1,,..., x n ; ω, R) R>0 δ 0 (x 1,..., x n ) = lim sup ω 0 δ 0 (x 1,,..., x n ; ω), where δ 0 (x 1,,..., x n ) is called the free entropy dimension of x 1,..., x n. (Here we used an equivalent definition by Jung [12])

18 Proposition 4 Let R > max{ X 1,..., X n } be a positive number. Then δ 0 (x 1,..., x n ) = lim sup ω 0 δ 0 (x 1,,..., x n ; ω, R).

19 Theorem 2 Let L(F n ) be the free group factor on n generators with the tracial state τ, and u 1,..., u n be the standard generators of L(F n ). Then δ 0 (u 1,..., u n ) = n. Proof: It follows from Proposition 2 that, for every m 1 and ɛ > 0, there are some positive integer k m,ɛ and a sequence of subsets {Ω m,ɛ (k)} k=k m,ɛ such that µ k (Ω m,ɛ (k)) 1 2, for k k m,ɛ, where µ k U(k) n. is normalized Haar measure on For each R > 1 and m 1, ɛ > 0, it is not hard to verify that, when m is large enough and ɛ is small enough, for any k k m,ɛ, Ω m,ɛ (k) Γ R (u 1,..., u n ; k, m, ɛ )

20 Note there exists constant C (not depending on k) such that for any ball centered at (U 1,..., U n ) with radius ω (with respect to 2-norm) we have µ k (Ball((U 1,..., U n ), ω)) (Cω) nk2, 0 < ω < 1. Thus, ν 2 (Γ R (u 1,..., u n ; k, m, ɛ ), ω) (Cω) nk2. Therefore δ 0 (u 1,..., u n ) = n.

21 Lemma 1 Let x be a normal element in a von Neumann algebra M with a tracial state τ. Let R > x. For every ω > 0, there is some positive integer m such that, for all k 1, if A, B are two matrices in M k (C) satisfying A, B Γ R (x; k, m, 1 m ), then there is some unitary matrix U in U(k) such that UAU B 2 ω. Proof: Suppose on the contrary that the following holds: there is some ω 0 > 0 such that for every m 1, there is some k m 1 and some self-adjoint matrices A m, B m in M km (C) satisfying A m, B m Γ R (x; k, m, 1 m ),

22 and UA m U B m 2 > ω 0 for all unitary matrix U in U(k m ). Let α be a free filter on N then denote by M km (C) α the ultrapower of {M km (C)} m=1 along the free filter α, (M km ) γ is the quotient of the C algebra m M km (C) by the 0-ideal of the norm α, where (Y m ) m=1 2,α = lim m α Y m 2 for each (Y m ) m=1 in m M km (C). Let ψ or φ be the mapping from W (x) to W ([(A m ) m=1 ]), or W ([(B m ) m=1 ]), induced by x [(A m ) m=1 ] x [(B m ) m=1 ] Then ψ and φ are two trace preserving - isomorphisms.

23 For any ω > 0, there are mutually orthogonal projections p 1,..., p n in W (x) and complex numbers a 1,..., a n such that x n i=1 a i p i 2 ω/3. Thus [(A m ) m=1 ] [(B m ) m=1 ] n i=1 n i=1 a i ψ(p i ) 2,α ω/3 a i φ(p i ) 2,α ω/3 Note that M km (C) α is a factor of type II 1. There is a sequence of unitary matrices {U m } m=1 with U m in M km (C) such that [(U m ) m=1 ]( n i=1 a i ψ(p i ))[(U m ) m=1 ] = n i=1 a i φ(p i ).

24 Hence lim m α U m A m U m B m 2 ω, which contradicts with the assumption that UA m U B m 2 > ω 0 for all unitary matrix U in U(k m ). Therefore, the statement of the lemma is true. Theorem 3 Suppose that x is a normal element in a von Neumann algebra M with a faithful normal tracial state τ. Then δ 0 (x) 1.

25 2.4 Some results 1. Voiculescu showed that if x is an selfadjoint element of a von Neumann algebra M with the tracial state τ, then δ 0 (x) = 1 t (τ(e({t})) 2, where E is the spectral prejection of x in (M, τ). 2. Jung showed that if M is a finite hyperfinite von Neumann algebra, i.e. M M 0 ( N i=1 M n i (C)) with a faithful tracial state τ = t 0 τ 0 ( N i=1 t iτ ni ),

26 where M 0 is a diffuse von Neumann algebra. Then δ 0 (x 1,..., x n ) = 1 N i=1 t 2 i n 2 i. 3. Voiculescu showed that if u 1,..., u n are the standard generators of L(F n ), then δ 0 (u 1,..., u n ) = n. 4. Voiculesu showed that if X 1,..., X n is a family of generators of a finite von Neumann algebra M with a Cartan subalgebra, then δ 0 (X 1,..., X n ) Ge showed that if X 1,..., X n is a family of generators of nonprime II 1 factor M,

27 i.e., M is a tensor product of two type II 1 factors, then δ 0 (X 1,..., X n ) 1. In particular, L(F(n))(n > 1) is prime. 6. K. Dykema computed the free entropy dimension for the von Neumann algebras with finite multiplicity and the ones with property C. 7. M. Stefan showed that the free group factors L(F(n)) don t have nonprime subfactors with finite index.

28 8. Let M be a type II 1 von Neumann algebra. If there is a sequence of Haar unitaries (a unitary u is called the Haar unitary if τ(u n ) = 0 for n 0){u j } j=1 in M such that (a) {u j } j=1 generate M, and (b) u j+1 u j u j+1 is in the von Neumann subalgebra generated by {u 1,..., u j } for all j 1, then, we showed δ 0 (X 1,..., X n ) 1, when X 1,..., X n is a family of generators of M. 9. Jung and Shlyakhtenko showed that all generating set of all property T von

29 Neumann algebra have free entropy dimension lass than or equal to 1.

30 Topological free entropy dimension Covering number in complex matrix algebras M k (C) Let M k (C) be the k k full matrix algebra with entries in C, and τ k be the normalized trace on M k (C), i.e., τ k = 1 kt r, where T r is the usual trace on M k (C). Let M k (C) n denote the direct sum of n copies of M k (C). Let M s.a k (C) be the subalgebra of M k (C) consisting of all self-adjoint matrices of M k (C). Let (M s.a k (C))n be the direct sum of n copies of M s.a k (C).

31 Let denote the operator norm on M k (C) n, i.e., (A 1,..., A n ) = max{ A 1,..., A n } for all (A 1,..., A n ) in M k (C) n. For every ω > 0, we define the ω- -ball Ball(B 1,..., B n ; ω, ) centered at (B 1,..., B n ) in M k (C) n to be the subset of M k (C) n consisting of all (A 1,..., A n ) in M k (C) n such that (A 1,..., A n ) (B 1,..., B n ) < ω. Definition 2 Suppose that Σ is a set in M k (C) n. We define ν (Σ, ω) to be the minimal number of ω- -balls that consist a covering of Σ in M k (C) n.

32 Noncommutative polynomials In this article, we always assume that A is a C - algebra. Let x 1,..., x n, y 1,..., y m are selfadjoint elements in A. Let 1 C X 1,..., X n, Y 1,..., Y m be the noncommutative polynomials in the undeteminates X 1,..., X n, Y 1,..., Y m. Let {P r } r=1 be the collection of all noncommutative polynomials in C X 1,..., X n, Y 1,..., Y m with rational coefficients. (Here rational coefficients means that the real and imaginary parts of all coefficients of P r are rational numbers).

33 Voiculescu s Norm-microstates For all integers r, k 1, real numbers R, ɛ > 0 and noncommutative polynomials P 1,..., P r, we define Γ R (x 1,..., x n, y 1,..., y m ; k, ɛ, P 1,..., P r ) to be the subset of (M s.a k (C))n+m consisting of all these (A 1,..., A n, B 1,..., B m ) (M s.a k (C))n+m satisfying max{ A 1,..., A n, B 1,..., B m } R and P j (A 1,..., A n, B 1,..., B m ) P j (x 1,..., x n, y 1,..., y m ) ɛ, 1 j r.

34 In the definition of norm-microstates space, we use the following assumption. If P j (x 1,..., x n, y 1,..., y m ) = α 0 I A + N s=1 1 i 1,...,i s n+m α i1 i s z i1 z is where z 1,..., z n+m denotes x 1,..., x n, y 1,..., y m and α 0, α i1 i s are in C, then P j (A 1,..., A n, B 1,..., B m ) = α 0 I k + N s=1 1 i 1,...,i s n+m α i1 i s Z i1 Z is where Z 1,..., Z n+m denotes A 1,..., A n, B 1,..., B m and I k is the identity matrix in M k (C).

35 We define the norm-microstates of x 1,..., x n in the presence of y 1,..., y m, denoted by Γ R (x 1,..., x n : y 1,..., y m ; k, ɛ, P 1,..., P r ) as the projection of Γ R (x 1,..., x n, y 1,..., y m ; k, ɛ, P 1,..., P r ) onto the space (M s.a k (C))n via the mapping (A 1,..., A n, B 1,..., B m ) (A 1,..., A n ). Voiculescu s topological entropy dimension We define ν (Γ R (x 1,..., x n : y 1,..., y m ; k, ɛ, P 1,..., P r ), ω) to be the covering number of the set Γ R (x 1,..., x n : y 1,..., y m ; k, ɛ, P 1,..., P r )

36 by ω- -balls in the metric space (M s.a k (C))n equipped with operator norm. Define δ top (x 1,..., x n, y 1,..., y m ; r, ɛ, R, ω) = ν (Γ lim sup R ( x, y; k, ɛ, P 1,..., P r ), ω) k k 2 log ω δ top (x 1,..., x n, y 1,..., y m ; ω) = sup R inf r,ɛ δ top (x 1,..., x n, y 1,..., y m ; r, ɛ, R, ω) The topological entropy dimension of x 1,..., x n in the presence of y 1,..., y m is defined by δ top (x 1,..., x n, y 1,..., y m ) = lim sup ω δ top(x 1,..., x n, y 1,..., y m ; ω)

37 Results of topological free entropy dimension 1. Suppose that x 1,..., x n is a free family of semicircular elements. Voiculescu showed, using the result by Haagerup and Thorbjornsen, that δ top (x 1,..., x n ) = n. 2. Suppose that x is a self-adjoint element in a unital C algebra A. Then δ top (x) = 1 1 n, where n is the cardinality of the spectrum of x in A. 3. Suppose that A is a finite dimensional C algebra and dim C A is the complex

38 dimension of A. If x 1,..., x n is a family of self-adjoint generators of A, then δ top (x 1,..., x n ) = 1 1 dim C A. 4. Suppose that A is a C algebra with a unique tracial state τ. Then δ top (x 1,..., x n ) δ 0 (x 1,..., x n : τ), for any family of self-adjoint generators x 1,..., x n of A. 5. Suppose that A is a infinite dimensional, unital, simple C algebra with a unique tracial state and suppose that A has the approximation property. Then δ top (x 1,..., x n ) 1,

39 where x 1,..., x n is a family of self-adjoint generators of A. Suppose A is a C algebra and x 1,..., x n is a family of self-adjoint elements of A that generates A as a C algebra. If for any R > max{ x 1,..., x n, y 1,..., y m }, r > 0, ɛ > 0, there is a sequence of positive integers k 1 < k 2 < such that Γ (top) R ( x : y; k s, ɛ, P 1,..., P r ), s 1 then A is called having approximation property. 6. We have δ top (x 1,..., x n ) = 1 where x 1,..., x n is a family of self-adjoint generators of UFH algebra, or irrational rotation algebra, or C red (F 2 ) min C red (F 2 ).

40 7. Suppose that A and B are two unital C algebras and x 1 y 1,..., x n y n is a family of self-adjoint elements that generates A B. Assume and s = δ top (x 1,..., x n ) t = δ top (y 1,..., y n ). (i) If s 1 or t 1, then δ top (x 1 y 1,..., x n y n ) = max{s, t} (ii) If s < 1, t < 1 and both families {x 1,..., x n }, {y 1,..., y n } are stable, then δ top (x 1 y 1,..., x n y n ) = st 1 s + t 2 ; and the family of elements x 1 y 1,..., x n y n is also stable.

41 A family of elements x 1,..., x n in A is called stable if for any α < δ top (x 1,..., x n ) there are positive numbers C 3 > 0 and ω 0 > 0, r 0 1, k 0 1 so that ν (Γ (top) R (x 1,..., x n ; q k 0, 1 r, P 1,..., P r ), ω) C (q k 0) 2 3 ω 0 < ω < ω 0, r > r 0, q N. ( ) 1 α (q k0 ) 2, 8. (Work in progress) Suppose that A a unital C algebras and B = A M n (C). Suppose x 1 = x (1) st e st,..., x n = x (n) st e st is a family of self-adjoint elements in B that generates B, where {e st } n st=1 is the canonical system of matrix units of

42 M n (C). Then δ top (x 1,..., x n ) = 1 + δ top( x (i) st ) 1 n 2. Reference 1. A. Connes, A factor of type II 1 with countable fundamental group, J. Operator Theory 4 (1980), A. Connes and V. Jones, Property T for von Neumann algebras, Bull. London Math. Soc., 17 (1985), Kenneth J Dykema, Two applications of free entropy, Math. Ann. 308 (1997), no. 3,

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45 13. M. Stefan The primality of subfactors of finite index in the interpolated free group factors, Proc. of A.M.S. vol 126, no. 8, D. Voiculescu, The analogues of entropy and of Fisher s information measure in free probability theory II, Invent. Math., 118 (1994), D. Voiculescu, The analogues of entropy and of Fisher s information measure in free probability theory III: The absence of Cartan subalgebras, Geom. Funct. Anal. 6 (1996) D. Voiculescu, Free entropy dimension 1 for some generators of property T

46 factors of type II 1, preprint UC Berkeley, RAM-753, Feb D. Voiculescu, K. Dykema and A. Nica, Free Random Variables, CRM Monograph Series, vol. 1, AMS, Providence, R.I., 1992.