Quantum Exchangeable Sequences of Algebras

Quantum Rotatability

In arXiv:0807.0677, Köstler and Speicher observed that de Finetti's theorem on exchangeable sequences has a free analogue if one replaces exchangeability by the stronger condition of invariance under quantum permutations. In this paper we study sequences of noncommutative random variables whose joint distribution is invariant under quantum orthogonal transformations. We prove a free analogue of Freedman's characterization of conditionally independent Gaussian families, namely an infinite sequence of self-adjoint random variables is quantum orthogonally invariant if and only if they form an ope…

Failure of the Ryll-Nardzewski theorem on the CAR algebra

Spreadability of a sequence of random variables is a distributional symmetry that is implemented by suitable actions of $\mathbb{J}_\mathbb{Z}$, the unital semigroup of strictly increasing maps on $\mathbb{Z}$ with cofinite range. We show that $\mathbb{J}_\mathbb{Z}$ is left amenable but not right amenable, although it does admit a right Folner sequence. This enables us to prove that on the CAR algebra ${\rm CAR}(\mathbb{Z})$ there exist spreadable states that fail to be exchangeable. Moreover, we also show that on ${\rm CAR}(\mathbb{Z})$there exist stationary states that fail to be spreadable…

A noncommutative De Finetti theorem for boolean independence

We introduce a family of quantum semigroups and their natural coactions on noncommutative polynomials. We present three invariance conditions, associated with these coactions, for the joint distribution of sequences of selfadjoint noncommutative random variables. For one of the invariance conditions, we prove that the joint distribution of an infinite sequence of noncommutative random variables satisfies it is equivalent to the fact that the sequence of the random variables are identically distributed and boolean independent with respect to the conditional expectation onto its tail algebra. Th…

A noncommutative extended de Finetti theorem

The extended de Finetti theorem characterizes exchangeable infinite random sequences as conditionally i.i.d. and shows that the apparently weaker distributional symmetry of spreadability is equivalent to exchangeability. Our main result is a noncommutative version of this theorem. In contrast to the classical result of Ryll-Nadzewski, exchangeability turns out to be stronger than spreadability for infinite noncommutative random sequences. Out of our investigations emerges noncommutative conditional independence in terms of a von Neumann algebraic structure closely related to Popa's notion of c…

De Finetti theorem on the CAR algebra

The symmetric states on a quasi local C*-algebra on the infinite set of indices J are those invariant under the action of the group of the permutations moving only a finite, but arbitrary, number of elements of J. The celebrated De Finetti Theorem describes the structure of the symmetric states (i.e. exchangeable probability measures) in classical probability. In the present paper we extend De Finetti Theorem to the case of the CAR algebra, that is for physical systems describing Fermions. Namely, after showing that a symmetric state is automatically even under the natural action of the parity…

When central sequence C*-algebras have characters

We investigate C*-algebras whose central sequence algebra has no characters, and we raise the question if such C*-algebras necessarily must absorb the Jiang-Su algebra (provided that they also are separable). We relate this question to a question of Dadarlat and Toms if the Jiang-Su algebra always embeds into the infinite tensor power of any unital C*-algebra without characters. We show that absence of characters of the central sequence algebra implies that the C*-algebra has the so-called strong Corona Factorization Property, and we use this result to exhibit simple nuclear separable unital C…

中心序列代数的二分性

A dichotomy for central sequence algebras

本文证明了可分C*-代数的中心序列代数必为次齐性或非正合，证实Enders–Shulman猜想，并将该二分性推广到其他大规模C*-代数。

Central Sequences in Subhomogeneous Unital C*-algebras

Suppose A is a unital subhomogeneous C*-algebra. We show that every central sequence in A is hypercentral if and only if every pointwise limit of a sequence of irreducible representations is multiplicity free. We also show that every central sequence in A is trivial if and only if every pointwise limit of irreducible representations is irreducible. We also give a nice repesentation of the latter algebras.

Non-commutativity of the central sequence algebra for separable non-type I C$^{\ast}$-algebras

We show that if $A$ is a separable, simple and non-type I C$^{\ast}$ algebra, then for every properly infinite hyperfinite von Neumann algebra $M$ with separable predual, its Ocneanu ultrapower $M'\cap M^ω$ arises as a sub-quotient of the central sequence algebra $F(A)$ defined by the second named author. In particular, this answers affirmatively the question of the second named author (Abel Symposium '04): the central sequence C$^{\ast}$-algebra of the reduced free group C$^{\ast}$-algebra $C_{\rm{red}}^*(\mathbb{F}_2)$ is non-commutative.

II$_1$ factors with exotic central sequence algebras

We provide a class of separable II$_1$ factors $M$ whose central sequence algebra is not the "tail" algebra associated to any decreasing sequence of von Neumann subalgebras of $M$. This settles a question of McDuff \cite{Mc69d}.

Commutativity of central sequence algebras

The question of which separable C*-algebras have abelian central sequence algebras was raised and studied by Phillips ([Ph88]) and Ando-Kirchberg ([AK14]). In this paper we give a complete answer to their question: A separable C*-algebra $A$ has abelian central sequence algebra if and only if A satisfies Fell's condition. Moreover, we introduce a higher-dimensional analogue of Fell's condition and show that it completely characterizes subhomogeneity of central sequence algebras. In contrast, we show that any non-trivial extension by compact operators has not only non-abelian but not even resid…