Inclusions of C*-algebras arising from fixed-point algebras
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We examine inclusions of C *-algebras of the form AH ⊆ A Ìr G, where G and H are groups acting on a unital simple C *-algebra A by outer automorphisms and H is finite. It follows from a theorem of Izumi that AH ⊆ A is C *-irreducible, in the sense that all intermediate C *-algebras are simple. We show that AH ⊆ A Ìr G is C *-irreducible for all G and H as above if and only if G and H have trivial intersection in the outer automorphisms of A, and we give a Galois type classification of all intermediate C *-algebras in the case when H is abelian and the two actions of G and H on A commute. We illustrate these results with examples of outer group actions on the irrational rotation C *-algebras. We exhibit, among other examples, C *-irreducible inclusions of AF-algebras that have intermediate C *-algebras that are not AF-algebras; in fact, the irrational rotation C *-algebra appears as an intermediate C *-algebra.
Originalsprog | Engelsk |
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Tidsskrift | Groups, Geometry, and Dynamics |
Vol/bind | 18 |
Udgave nummer | 1 |
Sider (fra-til) | 127-145 |
ISSN | 1661-7207 |
DOI | |
Status | Udgivet - 2024 |
Bibliografisk note
Funding Information:
Funding. Siegfried Echterhoff funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Project-ID 427320536 SFB 1442 and under Germany’s Excellence Strategy EXC 2044 390685587, Mathematics Münster: Dynamics, Geometry, Structure. Mikael Rørdam supported by a research grant from the Danish Council for Independent Research, Natural Sciences.
Publisher Copyright:
© 2023 European Mathematical Society.
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