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Essential Commutants of Semicrossed Products

Published online by Cambridge University Press:  20 November 2018

Kei Hasegawa
Kei Hasegawa*
Affiliation:
Graduate School of Mathematics, Kyushu University, Fukuoka 819-0395, Japan. e-mail: ma213034@math.kyushu-u.ac.jp
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Abstract

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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.

Keywords

47L6547A55essential commutantsemicrossed product
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 1 , 01 March 2015 , pp. 91 - 104
Copyright
Copyright © Canadian Mathematical Society 2015

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