Plan 1 Motivation & Terminology 2 Noncommutative De Finetti Theorems 3 Braidability 4 Quantum Exchangeability 5 References

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1 koestler University of Illinois at Urbana-Champaign / St. Lawrence University GPOTS 2008 University of Cincinnati June 18, 2008 [In parts joint work with Rolf Gohm and Roland Speicher]

2 Plan 1 Motivation & Terminology 2 Noncommutative De Finetti Theorems 3 Braidability 4 Quantum Exchangeability 5 References

3 Motivation Though many probabilistic symmetries are conceivable [...], four of them - stationarity, contractability, exchangeablity and rotatability - stand out as especially interesting and important in several ways: Their study leads to some deep structural theorems of great beauty and significance [...]. Olav Kallenberg (2005) Question: Can one transfer the related concepts to noncommutative probability theory and do they turn out to be fruitful in the study of the structure of operator algebras?

4 Hierarchy of distributional symmetries invariant objects stationary contractable exchangeable rotatable transformations shifts sub-sequences permutations isometries Topic of this talk: invariant objects are generated by an infinite sequence of random variables only the first three symmetries are considered contractable = spreadable

5 Motivating Example for De Finetti theorem Any exchangeable process is an average of i.i.d. processes. Theorem (De Finetti 1931) X 1, X 2,... infinite sequence of {0, 1}-valued random variables s.t. P(X 1 = e 1,..., X n = e n )=P(X π(1) = e 1,..., X π(n) = e n ) holds for all n N, permutations π and every e 1,..., e n {0, 1}. Then there exists a unique probability measure µ on [0, 1] such that P(X 1 = e 1,..., X n = e n )= p s (1 p) n s dµ(p), where s = e 1 + e e n.

6 Terminology of noncommutative probability Probability space (A,ϕ) A = von Neumann algebra with separable predual ϕ = faithful normal state on A Random variable ι: (A 0,ϕ 0 ) (A,ϕ) ι: A 0 A injective *-homomorphism satisfying ι(1l A0 )=1l A, ϕ ι = ϕ 0 and ισ ϕ 0 t = σ ϕ t ι Automorphisms of (A,ϕ) Aut(A,ϕ) = *-automorphisms α of A with ϕ α = ϕ Remark For commutative algebra of functions we get classical random variables and measure preserving transformations.

7 Noncommutative independence and commuting squares Given the probability space (A,ϕ), let A 0 A 1, A 2 be three von Neumann subalgebras of A such that the ϕ-preserving conditional expectations E i : A A i exist (i =1, 2, 3). Then A 1 and A 2 are said to be A 0 -independent if one of the following (equivalent) conditions is satisfied: 1 E 0 (xy) =E 0 (x)e 0 (y) for all x A 1 and y A 2 2 E 1 E 2 = E 0 3 the diagram A 1 A is a commuting square A 0 A 2 Remark For commutative algebras of functions on a probability space we obtain conditional independence with respect to a sub-σ-algebra.

8 Conditional Independence of random sequences Let I, J N 0. A sequence of random variables, for short: random sequence, ι (ι n ) n 0 :(A 0,ϕ 0 ) (A,ϕ) is B-independent if {ιi (A 0 ) i I } B and {ιj (A 0 ) j J} B are B-independent whenever I J =. Remark B may not be contained in {ι i (A 0 ) i I }

9 Noncommutative distributions Two random sequences ι (ι n ) n 0 and ι ( ι n ) n 0 from (A 0,ϕ 0 ) to (A,ϕ) have the same noncommutative distribution if ϕ ( ι i(1) (a 1 ) ι i(2) (a 2 ) ι i(n) (a n ) ) ϕ ( ι i(1) (a 1 ) ι i(2) (a 2 ) ι i(n) (a n ) ) for all n-tuples i: {1, 2,..., n} N 0,(a 1,..., a n ) A n 0 and n N. Notation (ι 0,ι 1,ι 2,...) distr = ( ι 0, ι 1, ι 2,...)

10 Noncommutative distributional symmetries Just as in the classical case we can now talk about distributional symmetries. A random sequence ι (ι n ) n 0 is exchangeable if (ι 0,ι 1,ι 2,...) distr = (ι π(0),ι π(1),ι π(2),...) for any finite permutation π S of N 0. spreadable if (ι 0,ι 1,ι 2,...) distr = (ι n0,ι n1,ι n2,...) for any subsequence (n 0, n 1, n 2,...) of (0, 1, 2,...). stationary if (ι 0,ι 1,ι 2,...) distr = (ι k,ι k+1,ι k+2,...) for all k N. Lemma (Hierarchy of distributional symmetries) Exchangeability Spreadability Stationarity

11 Classical dual version of extended De Finetti theorem Let ι (ι n ) n 0 :(A 0,ϕ 0 ) (A,ϕ) be a random sequence with A tail := ι k (A 0 ) n 0 and consider: (a) ι is exchangeable (c) ι is spreadable (e) ι is A tail -independent k n Theorem (De Finetti 31, Ryll-Nardzewski 57,..., K. 08) A L (A, Σ,µ) implies: (a) (c) (e) Remark Idea of dual version seems to be new!

12 Noncommutative De Finetti theorems with commutativity conditions Let ι (ι n ) n 0 :(A 0,ϕ 0 ) (A,ϕ) be a random sequence with A tail := ι k (A 0 ) n 0 and consider: (a) ι is exchangeable (e) ι is A tail -independent Theorem (Størmer 69,..., K. 08) Ranges of ι n s commute: k n (a) (e) Theorem (Accardi & Lu 93,..., K. 08) A tail is abelian: (a) (e)

13 Noncommutative extended De Finetti theorem Let ι (ι n ) n 0 :(A 0,ϕ 0 ) (A,ϕ) be a random sequence with A tail := ι k (A 0 ) n 0 and consider: (a) ι is exchangeable (c) ι is spreadable k n (d) ι is stationary and A tail -independent (e) ι is A tail -independent Theorem (K ) (a) (c) (d) (e), but (a) (c) (d) (e) Remark Spreadability implies conditional independence is the hard part!

14 Artin braid groups B n Algebraic Definition (Artin 1925) B n is presented by n 1 generators σ 1,..., σ n 1 satisfying σ i σ j σ i = σ j σ i σ j if i j = 1 (B1) σ i σ j = σ j σ i if i j > 1 (B2) 0 1 i-1 i 0 1 i-1 i Figure: Artin generators σ i (left) and σ 1 i (right) B 1 B 2 B 3... B (inductive limit)

15 Braidability Definition (Gohm & K. 08) A random sequence ι (ι n ) n 0 :(A 0,ϕ 0 ) (A,ϕ) is braidable if there exists a representation ρ: B Aut(A,ϕ) satisfying: ι n = ρ(σ n σ n 1 σ 1 )ι 0 for all n 1; ι 0 = ρ(σ n )ι 0 if n 2. Braidability extends exchangeability If ρ(σ 2 n) = id for all n, one has a representation of S. ι is exchangeable ι is braidable with ρ(σ 2 n) = id for all n.

16 Braidability implies spreadability It turns out that we can insert braidability between exchangeability and spreadability in the noncommutative de Finetti theorem and obtain a large and interesting class of spreadable sequences in this way. Consider the conditions: (a) ι is exchangeable (b) ι is braidable (c) ι is spreadable (d) ι is stationary and A tail -independent Theorem (Gohm & K. 08) (a) (b) (c) (d), but (a) (b) (d) Open Problem Construct spreadable sequence which fails to be braidable.

17 Wang s quantum permutation group (1998) The quantum permutation group A s (n) is the universal unital C*-algebra generated by elements {u ij i, j =1,... n} such that 1 u ij = u ij = u 2 ij (orthogonal projections) 2 the elements of each row and each column of the matrix u 11 u 12 u 1n u 21 u 22 u 2n... u n1 u nn Remark form a partition of unity, i.e. are orthogonal and sum up to 1. A s (n) is a compact quantum group in the sense of Woronowicz

18 Quantum Exchangeability Given a probability space (A,ϕ) with an infinite sequence x 1, x 2,... A, define the action of A s (k) on the tuple (x 1, x 2,..., x k ) by x i k u ij x j A s (k) A. i=1 Definition (K. & Speicher 08) The distribution of (x 1, x 2,...) is quantum exchangeable if ϕ ( ) k x i(1) x i(n) = u i(1)j(1) u i(n)j(n) ϕ ( ) x j(1) x j(n) j(1),...,j(n)=1 for all 1 i(1),... i(n) k and all k, n N.

19 A new characterization of freeness with amalgamation Theorem (K. & Speicher 08) Let (A,ϕ) be a probability space and consider an infinite sequence x 1, x 2,... A. Then the following two statements are equivalent: (a) (x n ) n N is quantum exchangeable (b) (x n ) n N is identically distributed and free over A tail Corollary (K. & Speicher 08 Suppose (x n ) n N is quantum exchangeable. If A tail is trivial, then (x n ) n N is free.

20 References. Noncommutative independence from the braid group B. (With Rolf Gohm) On Lehner s free noncommutative analogue of the de Finetti theorem. A noncommutative de Finetti theorem: Invariance under quantum permutations is equivalent to freeness with amalgamation. (With Roland Speicher) Preprint 2008, to be posted at arxiv.org