An overview of groupoid crossed products in dynamical systems. (English) Zbl 1367.46054
Martinetti, Pierre (ed.) et al., Noncommutative geometry and optimal transport. Workshop on noncommutative geometry and optimal transport, Besançon, France, November 27, 2014. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2297-4/pbk; 978-1-4704-3560-8/ebook). Contemporary Mathematics 676, 211-223 (2016).
Summary: The interaction between dynamical systems and crossed products provides a mechanism for studying group actions using tools from \(C^*\)-algebras. There are generalizations of these ideas in the context of groupoids. A main result in this setting is an extension of a fundamental theorem of Green, which presents a Morita equivalence between a crossed product involving the action of a group and the \(C^*\)-algebra of the corresponding quotient space. Another property is the Renault’s equivalence theorem, which also provides a Morita equivalence between crossed products of dynamical systems defined as the action of groupoids on \(C^*\)-bundles. These results highlight a deep connection between groupoids \(C^*\)-algebras and dynamical systems. In this paper, we provide a short expository overview listing the main definitions and properties required to understand these results.
For the entire collection see [Zbl 1353.46001].
For the entire collection see [Zbl 1353.46001].
MSC:
| 46L55 | Noncommutative dynamical systems |
| 46L05 | General theory of \(C^*\)-algebras |
| 22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |
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