Essential Commutants of Semicrossed Products

Abstract

AbstractLet α: G ↷ 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-1S. 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) = (ℤ, ℤ+), which extends a famous result due to Davidson (1977) for the classical analytic Toeplitz operators.