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An equivariant Brauer semigroup and the symmetric imprimitivity theorem


Authors: Astrid an Huef, Iain Raeburn and Dana P. Williams
Journal: Trans. Amer. Math. Soc. 352 (2000), 4759-4787
MSC (2000): Primary 46L05, 46L35
DOI: https://doi.org/10.1090/S0002-9947-00-02618-0
Published electronically: June 14, 2000
MathSciNet review: 1709774
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Abstract: Suppose that $(X,G)$ is a second countable locally compact transformation group. We let $\operatorname{S}_G(X)$ denote the set of Morita equivalence classes of separable dynamical systems $(A,G,\alpha)$ where $A$ is a $C_{0}(X)$-algebra and $\alpha$ is compatible with the given $G$-action on $X$. We prove that $\operatorname{S}_{G}(X)$ is a commutative semigroup with identity with respect to the binary operation $[A,G,\alpha][B,G,\beta]=[A\otimes_{X}B,G,\alpha\otimes_{X}\beta]$ for an appropriately defined balanced tensor product on $C_{0}(X)$-algebras. If $G$and $H$ act freely and properly on the left and right of a space $X$, then we prove that $\operatorname{S}_{G}(X/H)$ and $\operatorname{S}_{H}(G\backslash X)$ are isomorphic as semigroups. If the isomorphism maps the class of $(A,G,\alpha)$to the class of $(B,H,\beta)$, then $A\rtimes_{\alpha}G$ is Morita equivalent to $B\rtimes_{\beta}H$.


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Additional Information

Astrid an Huef
Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755-3551
Address at time of publication: Department of Mathematics, University of Denver, Denver, Colorado 80208
Email: astrid@cs.du.edu

Iain Raeburn
Affiliation: Department of Mathematics, University of Newcastle, Callaghan, New South Wales 2308, Australia
Email: iain@math.newcastle.edu.au

Dana P. Williams
Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755-3551
Email: dana.williams@dartmouth.edu

DOI: https://doi.org/10.1090/S0002-9947-00-02618-0
Received by editor(s): November 25, 1998
Published electronically: June 14, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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