Essential Commutants of Semicrossed Products

Hasegawa, Kei

Canadian mathematical bulletin, Tome 58 (2015) no. 1, pp. 91-104

Résumé

Let $\alpha :\,G\,\curvearrowright \,M$ be a spatial action of a countable abelian group on a “spatial” von Neumann algebra $M$ and let $S$ be its unital subsemigroup with $G\,=\,{{S}^{-1}}S$ . We explicitly compute the essential commutant and the essential fixed-points, modulo the Schatten $p$ -class or the compact operators, of the ${{w}^{*}}$ -semicrossed product of $M$ by $S$ when ${{M}^{'}}$ contains no non-zero compact operators. We also prove a weaker result when $M$ is a von Neumann algebra on a finite dimensional Hilbert space and $\left( G,\,S \right)\,=\,\left( \mathbb{Z},\,{{\mathbb{Z}}_{+}} \right)$ , which extends a famous result due to Davidson (1977) for the classical analytic Toeplitz operators.

Détail

DOI : 10.4153/CMB-2014-057-x

Mots-clés : 47L65, 47A55, essential commutant, semicrossed product