Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1503.02460

Asymptotic structure of free product von Neumann algebras

Cyril Houdayer·Yoshimichi Ueda
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $(M, φ) = (M_1, φ_1) \ast (M_2, φ_2)$ be the free product of any $σ$-finite von Neumann algebras endowed with any faithful normal states. We show that whenever $Q \subset M$ is a von Neumann subalgebra with separable predual such that both $Q$ and $Q \cap M_1$ are the ranges of faithful normal conditional expectations and such that both the intersection $Q \cap M_1$ and the central sequence algebra $Q' \cap M^ω$ are diffuse (e.g. $Q$ is amenable), then $Q$ must sit inside $M_1$. This result generalizes the previous results of the first named author in [Ho14] and moreover completely settles the questions of maximal amenability and maximal property Gamma of the inclusion $M_1 \subset M$ in arbitrary free product von Neumann algebras.

Explore related subjects

math.OA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Cyril Houdayer, Yoshimichi Ueda. 2016-10-03. Asymptotic structure of free product von Neumann algebras. https://doi.org/10.1017/s0305004116000396

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Set-theoretic absoluteness for analysts and separable \cstar-algebras without Choice

Basic theory of separable C*-algebras can be developed without the Axiom of Choice, and it does not depend on the Continuum Hypothesis, Martin's Axiom, and other standard set-theoretic assumptions. This can be proved in two ways. First, by showing that the standard proofs do not require Choice. Second, by utilizing set-theoretic absoluteness theorems. We provide an introduction to projective complexity and absoluteness for analysts. We also give some limiting examples consistent with ZF, such as a commutative \cstar-algebra concretely represented on a Hilbert space but not isomorphic to $C(X)$ for any compact Hausdorff space $X$ and whose state space is not compact and has no extreme points.

math.OA↗
arXiv2026

Schur Multipliers with Unequal Operator and Completely Bounded Norms on $S_p$, $1<p\ne 2<\infty $

For every $1<p\neq2<\infty$, we exhibit an explicit Schur multiplier on $S_p$ whose operator norm is strictly smaller than its completely bounded norm. More precisely, for each such $p$, we construct a finitely supported Schur symbol $m_p$ such that \[ \|M_{m_p}\|_{p\to p} < \|M_{m_p}\|_{\mathrm{cb},p}. \] This answers the question raised by Lafforgue and de la Salle in 2011 after their Conjecture~1.8 and gives an affirmative answer to Statement~2 in Section~5 of Caspers and Wildschut (2019). In particular, it disproves Statement~3 of Caspers and Wildschut (2019) throughout the same range of exponents.

math.OA↗
arXiv2026

Stationary states on a $C^*$-algebra for an inner action

We study stationary states for actions of countable discrete groups on unital separable $C^*$-algebras. We prove that the stationary state space associated with an inner action of a subgroup of the unitary group that generates the algebra is a Choquet simplex. We also give a locality criterion covering Bernoulli shifts. The simplex structure yields a canonical decomposition of stationary states into tracial and purely nontracial parts. For inner actions, we characterize extreme stationary states by factoriality of their GNS von Neumann algebras. We further show that a stationary state is tracial if and only if its GNS von Neumann algebra is finite, and that it is purely nontracial if and only if this algebra is of type~$\mathrm{III}$. As applications, for $2\leq d\leq\infty$ the stationary state simplex of $C^*(\mathbb{F}_d)$ has a Poulsen face, while for a nontrivial property~$(T)$ group the stationary state simplex of $C^*(Γ)$ is not Poulsen. Finally, we give a sufficient spectral-gap condition for $S_μ(A)$ to be a Bauer simplex and construct a family $(A_d,Γ_d,μ_d)$ satisfying this condition.

math.OA↗