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