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Semicrossed products of operator algebras by semigroups

About this Title

Kenneth R. Davidson, Pure Mathematics Department, University of Waterloo, Waterloo, ON N2L–3G1, Canada, Adam H. Fuller, Mathematics Department, University of Nebraska-Lincoln, Lincoln, Nebraska 68588–0130 and Evgenios T. A. Kakariadis, Previous: Pure Mathematics Department, University of Waterloo, Waterloo, ON N2L–3G1, Canada \phantom{oo} Previous: Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva 84105, Israel \phantom{oo} Current: School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 247, Number 1168
ISBNs: 978-1-4704-2309-4 (print); 978-1-4704-3697-1 (online)
Published electronically: December 21, 2016
Keywords: Dynamical systems of operator algebras, semicrossed products, C*-envelope, C*-crossed products.
MSC: Primary 47A20, 47L25, 47L65, 46L07

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Table of Contents


  • 1. Introduction
  • 2. Preliminaries
  • 3. Semicrossed products by abelian semigroups
  • 4. Nica-covariant semicrosssed products
  • 5. Semicrossed products by non-abelian semigroups


We examine the semicrossed products of a semigroup action by $*$-endomorphisms on a C*-algebra, or more generally of an action on an arbitrary operator algebra by completely contractive endomorphisms. The choice of allowable representations affects the corresponding universal algebra. We seek quite general conditions which will allow us to show that the C*-envelope of the semicrossed product is (a full corner of) a crossed product of an auxiliary C*-algebra by a group action.

Our analysis concerns a case-by-case dilation theory on covariant pairs. In the process we determine the C*-envelope for various semicrossed products of (possibly nonselfadjoint) operator algebras by spanning cones and lattice-ordered abelian semigroups.

In particular, we show that the C*-envelope of the semicrossed product of C*-dynamical systems by doubly commuting representations of $\mathbb {Z}^n_+$ (by generally non-injective endomorphisms) is the full corner of a C*-crossed product. In consequence we connect the ideal structure of C*-covers to properties of the actions. In particular, when the system is classical, we show that the C*-envelope is simple if and only if the action is injective and minimal.

The dilation methods that we use may be applied to non-abelian semigroups. We identify the C*-envelope for actions of the free semigroup $\mathbb {F}_+^n$ by automorphisms in a concrete way, and for injective systems in a more abstract manner. We also deal with C*-dynamical systems over Ore semigroups when the appropriate covariance relation is considered.

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