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Crossed Products of Operator Algebras
About this Title
Elias G. Katsoulis, Department of Mathematics, East Carolina University, Greenville, NC 27858 and Christopher Ramsey, Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada
Publication: Memoirs of the American Mathematical Society
Publication Year:
2019; Volume 258, Number 1240
ISBNs: 978-1-4704-3545-5 (print); 978-1-4704-5071-7 (online)
DOI: https://doi.org/10.1090/memo/1240
Published electronically: February 21, 2019
Keywords: $\mathrm {C}^*$-correspondence,
crossed product,
Dirichlet algebra,
gauge action,
semi-Dirichlet algebra,
semisimple algebra,
operator algebra,
TAF algebra,
tensor algebra
MSC: Primary 47L65; Secondary 46L07, 46L08, 46L55, 47B49, 47L40
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries
- 3. Definitions and Fundamental Results
- 4. Maximal C$^*$-covers, Iterated Crossed Products and Takai Duality
- 5. Crossed Products and the Dirichlet Property
- 6. Crossed Products and Semisimplicity
- 7. The Crossed Product as the Tensor Algebra of a C$^*$-correspondence.
- 8. Concluding Remarks and Open Problems
Abstract
We study crossed products of arbitrary operator algebras by locally compact groups of completely isometric automorphisms. We develop an abstract theory that allows for generalizations of many of the fundamental results from the selfadjoint theory to our context. We complement our generic results with the detailed study of many important special cases. In particular we study crossed products of tensor algebras, triangular AF algebras and various associated C$^\ast$-algebras. We make contributions to the study of C$^\ast$-envelopes, semisimplicity, the semi-Dirichlet property, Takai duality and the Hao-Ng isomorphism problem. We also answer questions from the pertinent literature.- Beatriz Abadie, Takai duality for crossed products by Hilbert $C^*$-bimodules, J. Operator Theory 64 (2010), no.Β 1, 19β34. MR 2669425
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