Inclusions of C*-algebras arising from fixed-point algebras

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Standard

Inclusions of C*-algebras arising from fixed-point algebras. / Echterhoff, Siegfried; Rørdam, Mikael.

In: Groups, Geometry, and Dynamics, Vol. 18, No. 1, 2024, p. 127-145.

Research output: Contribution to journal › Journal article › Research › peer-review

Harvard

Echterhoff, S & Rørdam, M 2024, 'Inclusions of C*-algebras arising from fixed-point algebras', Groups, Geometry, and Dynamics, vol. 18, no. 1, pp. 127-145. https://doi.org/10.4171/GGD/743

APA

Echterhoff, S., & Rørdam, M. (2024). Inclusions of C*-algebras arising from fixed-point algebras. Groups, Geometry, and Dynamics, 18(1), 127-145. https://doi.org/10.4171/GGD/743

Vancouver

Echterhoff S, Rørdam M. Inclusions of C*-algebras arising from fixed-point algebras. Groups, Geometry, and Dynamics. 2024;18(1):127-145. https://doi.org/10.4171/GGD/743

Author

Echterhoff, Siegfried ; Rørdam, Mikael. / Inclusions of C*-algebras arising from fixed-point algebras. In: Groups, Geometry, and Dynamics. 2024 ; Vol. 18, No. 1. pp. 127-145.

Bibtex

@article{2f39ac08eeed47ca9dccd0e92e6b5e6d,
title = "Inclusions of C*-algebras arising from fixed-point algebras",
abstract = "We examine inclusions of C *-algebras of the form AH ⊆ A {\`I}r 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 {\`I}r 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.",
keywords = "crossed product, fixed-point algebra, irrational rotation algebra, Irreducible inclusion of C-algebras",
author = "Siegfried Echterhoff and Mikael R{\o}rdam",
note = "Publisher Copyright: {\textcopyright} 2023 European Mathematical Society.",
year = "2024",
doi = "10.4171/GGD/743",
language = "English",
volume = "18",
pages = "127--145",
journal = "Groups, Geometry, and Dynamics",
issn = "1661-7207",
publisher = "European Mathematical Society Publishing House",
number = "1",

}

RIS

TY - JOUR

T1 - Inclusions of C*-algebras arising from fixed-point algebras

AU - Echterhoff, Siegfried

AU - Rørdam, Mikael

N1 - Publisher Copyright: © 2023 European Mathematical Society.

PY - 2024

Y1 - 2024

N2 - 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.

AB - 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.

KW - crossed product

KW - fixed-point algebra

KW - irrational rotation algebra

KW - Irreducible inclusion of C-algebras

U2 - 10.4171/GGD/743

DO - 10.4171/GGD/743

M3 - Journal article

AN - SCOPUS:85186472223

VL - 18

SP - 127

EP - 145

JO - Groups, Geometry, and Dynamics

JF - Groups, Geometry, and Dynamics

SN - 1661-7207

IS - 1

ER -

ID: 385838180