Quantum Symmetric States on Free Product C -Algebras

Transcription

1 Quantum Symmetric States on Free Product C -Algebras School of Mathematical Sciences University College Cork Joint work with Ken Dykema & John Williams arxiv: WIMCS-LMS Conference Classifying Structures for Operator Algebras and Dynamical Systems Aberystwyth University September 2013

2 What is the free probability counterpart to the following semi-classical result? Theorem (Størmer 1969) Let A be a unital C -algebra with state space S(A). Then e SS(A) = { 1 ψ ψ S(A)}. Here SS(A) denotes the set of all symmetric states on 1 A. Remarks SS(A) is a Choquet simplex and SS(A) 1:1 M 1 (S(A)) Hewitt & Savage s result (1955) on symmetric measures on Cartesian products is recovered for A = C(X ), with X compact Hausdorff space.

3 Symmetric states Definition Given a unital C*-algebra A, a C*-algebraic probability space (B, φ) and unital *-homomorphisms λ i : A B (i N), the sequence (λ i ) i is said to be exchangeable w.r.t. φ or the state φ is said to be symmetric w.r.t. (λ i ) i if φ(λ i1 (a 1 ) λ in (a n )) = φ(λ σ(i1 )(a 1 ) λ σ(ik ))(a n )). for all n N, all a 1,..., a n A, all permutations σ (on N).

4 Discussion of Exchangeability Tensor independence: Størmer s result is about the special case B = 1 A, where λ i is the canonical embedding of A into the i-th tensor product factor. Exchangeability implies noncommutative independence in terms of Popa s commuting squares (K 2010, K & Gohm 2009). Exchangeability is too weak as a distributional symmetry to single out free independence among all these different forms of noncommutative independence! New idea (K & Speicher 2009): Strengthen exchangeability to quantum exchangeability: permutation group quantum permutation group

5 Quantum Permutation Groups Definition and Theorem (Wang 1998) The quantum permutation group A s (k) is the universal unital C -algebra generated by e ij (i, j = 1,..., k subject to the relations 1. eij 2 = e ij = eij for all i, j = 1,..., k e 11 e 1k 2. each column and row of is a partition of unity e k1 e kk A s (k) is a compact quantum group in the sense of Woronowicz, in particular a Hopf C*-algebra. The abelianization of A s (k) is C(S k ), the continuous functions on the symmetric group S k.

6 Quantum Exchangeability/Quantum Symmetric States Definition (K & Speicher 09 / Dykema & K & Williams 13) Given a unital C*-algebra A, a C*-algebraic probability space (B, φ) and unital *-homomorphisms λ i : A B (i N), the sequence (λ i ) i is said to be quantum exchangeable w.r.t. φ or the state φ is said to be quantum symmetric w.r.t. (λ i ) i if 1l As(k)φ ( λ i1 (a 1 ) λ in (a n ) ) = k j 1,...,j n=1 e i1 j 1 e inj n φ ( λ j1 (a 1 ) λ jn (a n ) ) for all n N, all i 1,..., i n {1,..., k}, all a 1,..., a n A, all k k-matrices (e ij ) ij satisfying defining relations for A s (k). Remark One has the coaction k j=1 e ij λ j : A A s (k) B

7 Quantum symmetric states arising from freeness Proposition Let D B be unital C -algebras with conditional expectation E : B D (i.e. a projection of norm one onto D). Suppose that λ i : A B (i N) are *-homomorphisms such that E λ i are the same for all i and λ i (A)) i=1 is free w.r.t. E. Let A = 1 A and λ = i=1λ i : A B be the resulting free product *-homomorphism. For a state ρ on D consider the state ψ = ρ E λ. Then ψ is quantum symmetric. [The proof uses Speicher s free D-valued cumulants and the defining properties of the projections e ij from Wang s quantum permutation group.]

8 Tail algebras of (quantum) symmetric states Let ψ be a state on A = 1 A. Passing to the GNS representation (H ψ, π ψ, Ω ψ ) of (A, ψ), put M ψ = π ψ (A), ψ = Ωψ, Ω ψ. The tail algebra (of (λ i ) i ) for ψ is the von Neumann subalgebra T ψ = W ( n=1 i=n ) π ψ (λ i (A)) M ψ. Proposition (Dykema & K & Williams 2013) Suppose ψ is symmetric. Then there exists a normal ψ-preserving conditional expectation E ψ from M ψ onto T ψ.

9 Quantum symmetric distributions imply freeness with amalgamation Theorem (Dykema & K & Williams) Let ψ be a quantum symmetric state on A = 1 A and put B i := W ( π ψ (λ i (A)) T ψ ). Then (B i ) i=1 is free with respect to E ψ. [The proof is modeled along the one of K & Speicher (2009), but starts now in a C*-algebraic setting and does not assume faithfulness of a state as required for W*-probability spaces.]

10 What tail algebras can appear? In Størmer s setting of symmetric states on 1 A only abelian tail algebras can arise. But in our setting of quantum symmetric states one has: Theorem (Dykema & K 2012) Let N be a countable generated von Neumann algebra. Then there exists a unital C*-algebra A and a quantum symmetric state ψ on A = 1 A such that T ψ N. Remark Størmer s approach does not use tail algebras; the machinery of ergodic decompositions of states and Choquet theory is available.

11 Description of quantum symmetric states QSS(A) For a unital C*-algebra A, let V(A) be the set (of all equivalence classes) of quintuples (B, D, E, σ, ρ) such that (i) B is a von Neumann algebra, (ii) D is a unital von Neumann subalgebra of B, (iii) E : B D is a normal conditional expectation onto D, (iv) σ : A B is a unital homomorphism, (v) ρ is a normal state on D, (vi) the GNS rep. of ρ E is a faithful rep. of B, (vii) B is generated as a von Neumann algebra by σ(a) D, (viii) D is the smallest unital von Neumann subalgebra of B that satisfies E ( d 0 σ(a 1 )d 1 σ(a n )d n ) D whenever n N, d 0,..., d n D and a 1,..., a n A.

12 Theorem (Dykema & K & Williams) There is a bijection V(A) QSS(A) that assigns to W = (B, D, E, σ, ρ) V(A) the state ψ = ψ W as follows. Let (M, Ẽ) = ( D) i=1(b, E) be the amalgamated free product of von Neumann algebras, let π i : A σ B i-th comp. M, π := i=1 π i : A M free product *-homomorphism and set ψ = ρ Ẽ π. Under this correspondence, the following identifications of objects and resulting constructions are naturally made: from GNS construction T ψ M ψ π ψ ψ Eψ from quintuple W D M π ρ Ẽ Ẽ

13 N-pure states on von Neumann algebras A state ψ on a C*-algebra is pure if whenever ρ is a state on this C*-algebra with tρ ψ for some 0 < t < 1, it follows already ρ = ψ. Proposition Let ρ be a normal state on the von Neumann algebra D. TFAE: (i) the support projection of ρ is a minimal projection in D, (ii) ρ is pure. To emphasize this support property, a normal pure state ρ on D is called an n-pure state. Remark Von Neumann algebras without discrete type I parts possesss no n-pure states, but such von Neumann algebras may appear as tail algebra for a quantum symmetric state.

14 Extreme quantum symmetric states Theorem (Dykema & K & Williams 2013) Let ψ QSS(A) and let W = (B, D, E, σ, ρ) be the quintuple corresponding to ψ (under the bijection as indicated in the previous theorem). ψ e QSS(A) ρ is a n-pure state on D Remarks - Though having an n-pure state is a restriction on the tail algebra and forces it to have a discrete type I part, the tail algebra can still be quite complicated. - For various examples of quantum symmetric states with exotic tail algebras see our preprint.

15 Central quantum symmetric states ZQSS(A) := {ψ QSS(A) T ψ Z(π ψ (A) )} TQSS(A) := {ψ QSS(A) ψ is tracial} ZTQSS(A) := ZQSS(A) TQSS(A) Theorem (Dykema & K & Williams) ZQSS(A) and ZTQSS(A) are compact, convex subsets of QSS(A) and both are Choquet simplices, whose extreme points are the free product states and free product tracial states, respectively: e ZQSS(A) = { 1 ψ ψ S(A)} e ZTQSS(A) = { 1 τ τ TS(A)} Remark ZQSS(A) is the part of QSS(A) which is in analogy to Størmer s result on symmetric states on the minimal tensor product 1 A.

16 What about TQQS(A)?!... as just announced for the 2014 AMS-MAA Joint Mathematics Meeting in Baltimore... Theorem (Dabrowski & Dykema & Mukjherjee & Williams) TQQS(A) is a Choquet simplex. Thank You for your attention!