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Essential Commutants of Semicrossed Products

  Published:2014-11-13
 Printed: Mar 2015
  • Kei Hasegawa,
    Graduate School of Mathematics, Kyushu University, Fukuoka 819-0395, Japan
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Abstract

Let $\alpha\colon G\curvearrowright M$ be a spatial action of countable abelian group on a "spatial" von Neumann algebra $M$ and $S$ 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 $(G,S)=(\mathbb{Z},\mathbb{Z}_+)$, which extends a famous result due to Davidson (1977) for the classical analytic Toeplitz operators.
Keywords: essential commutant, semicrossed product essential commutant, semicrossed product
MSC Classifications: 47L65, 47A55 show english descriptions Crossed product algebras (analytic crossed products)
Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15]
47L65 - Crossed product algebras (analytic crossed products)
47A55 - Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15]
 

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