On approximately inner automorphisms of certain crossed product 𝐶*-algebras

Dădărlat, Marius; Pasnicu, Cornel

doi:10.1090/S0002-9939-1990-1021897-4

On approximately inner automorphisms of certain crossed product $C^ *$-algebras
HTML articles powered by AMS MathViewer

by Marius Dădărlat and Cornel Pasnicu

Proc. Amer. Math. Soc. 110 (1990), 383-385

DOI: https://doi.org/10.1090/S0002-9939-1990-1021897-4

PDF | Request permission

Abstract:

Let $G$ be a compact connected topological group having a dense subgroup isomorphic to $\mathbf {Z}$. Let $C(G) \stackrel {\rtimes }{\propto } \mathbf {Z}$ be the crossed product $C^*$-algebra of $C(G)$ with $\mathbf {Z}$, where $\mathbf {Z}$ acts on $G$ by rotations. Automorphisms of $C(G) \stackrel {\rtimes }{\propto } \mathbf {Z}$ leaving invariant the canonical copy of $C(G)$ are shown to be approximately inner iff they act trivially on $K_1(C(G) \stackrel {\rtimes }{\propto } \mathbf {Z})$.

References

Bruce Blackadar, $K$-theory for operator algebras, Mathematical Sciences Research Institute Publications, vol. 5, Springer-Verlag, New York, 1986. MR 859867, DOI 10.1007/978-1-4613-9572-0
Berndt Brenken, Approximately inner automorphisms of the irrational rotation algebra, C. R. Math. Rep. Acad. Sci. Canada 7 (1985), no. 6, 363–368. MR 813959
M. Dădărlat and C. Pasnicu, On certain automorphisms of reduced crossed products with discrete groups, J. Operator Theory 23 (1990), no. 1, 179–193. MR 1054824

Abstract harmonic analysis

Gert K. Pedersen, $C^{\ast }$-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR 548006
M. Pimsner and D. Voiculescu, Exact sequences for $K$-groups and Ext-groups of certain cross-product $C^{\ast }$-algebras, J. Operator Theory 4 (1980), no. 1, 93–118. MR 587369
Wladimiro Scheffer, Maps between topological groups that are homotopic to homomorphisms, Proc. Amer. Math. Soc. 33 (1972), 562–567. MR 301130, DOI 10.1090/S0002-9939-1972-0301130-8
G. Zeller-Meier, Produits croisés d’une $C^{\ast }$-algèbre par un groupe d’automorphismes, J. Math. Pures Appl. (9) 47 (1968), 101–239 (French). MR 241994