Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/0602158

Strongly singular MASA's and mixing actions in finite von Neumann algebras

Paul Jolissaint·Yves Stalder
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $Γ$ be a countable group and let $Γ_0$ be an infinite abelian subgroup of $Γ$. We prove that if the pair $(Γ,Γ_0)$ satisfies some combinatorial condition called (SS), then the abelian subalgebra $A=L(Γ_0)$ is a singular MASA in $M=L(Γ)$ which satisfies a weakly mixing condition. If moreover it satisfies a stronger condition called (ST), then it provides a singular MASA with a strictly stronger mixing property. We describe families of examples of both types coming from free products, HNN extentions and semidirect products, and in particular we exhibit examples of singular MASA's that satisfy the weak mixing condition but not the strong mixing one.

Explore related subjects

math.OAmath.GR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Paul Jolissaint, Yves Stalder. 2008-01-16. Strongly singular MASA's and mixing actions in finite von Neumann algebras. https://arxiv.org/abs/math/0602158

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

KEEP EXPLORING

Related papers

arXiv2026

The e-Property for Quantum Markov Semigroups

Let $(\mathcal{T}_t)_{t\geqslant0}$ be a quantum Markov semigroup on a von Neumann algebra and let $\mathcal{A}$ be an invariant Feller $C^*$-algebra. We formulate the $e$-property on the full state space and show that its quantitative defect is equivalent, within an optimal factor two, to the Hausdorff measure of non-compactness of observable orbits. We develop Cesàro and Lip-norm criteria and a generalised Doeblin minorisation by a state-space contracting positive map, including finite-dimensional and measure-prepare factorisations. When every observable has the $e$-property, the Jacobs--de Leeuw--Glicksberg decomposition identifies the reversible Choi--Effros algebra and the norm-stable sector. Expected and decoherence-free subalgebras transfer the property under norm decay of the complementary dynamics, including infinite-dimensional ultracontractive and finite-index Pimsner--Popa regimes. Finally, we establish the full comparison hierarchy among ordinary, eventual, Cesàro and pointwise eventual equicontinuity notions, reinforced by matrix Jamison and homoclinic flow counterexamples. Projection-face criteria are treated in a companion paper.

math.OA↗
arXiv2026

Geometric Aspects of $C^*$-Extreme Points II

Building on the results of \cite{HR}, we investigate the relationship between \(C^*\)-extremality and linear extremality in the closed unit balls of general unital \(C^*\)-algebras. We show that, unlike in von Neumann algebras, these two notions of extremality need not coincide in general \(C^*\)-algebras. More precisely, we prove that the unilateral shift is a linear extreme point, but not a \(C^*\)-extreme point, of the closed unit ball of the Toeplitz \(C^*\)-algebra. We further establish that \(C^*\)-extremality implies linear extremality whenever the \(C^*\)-algebra admits a faithful unital representation on a separable Hilbert space. We also prove this implication for elements admitting a factorization $x=v|x|$, where $v$ is a partial isometry belonging to the algebra, and extend the result to suitable quotient \(C^*\)-algebras, including the Calkin algebra. Finally, we characterize the linear extreme points of the closed unit ball of unital \(C^*\)-subalgebras of \(B(H)\) containing the compact operators.

math.OA↗
arXiv2026

Every classifiable simple C*-algebra has a C*-diagonal

We construct C*-diagonals in all UCT Kirchberg algebras. Together with known constructions of C*-diagonals in the stably finite setting by Xin Li, this shows that every classifiable C*-algebra (in the sense of the Elliott classification programme) has a C*-diagonal. In fact, we show that every UCT Kirchberg algebra admits an untwisted principal étale groupoid model with a finite-dimensional unit space.

math.OA↗