Generalized Bunce–Deddens algebras

Orfanos, Stefanos

doi:10.1090/S0002-9939-09-10071-0

Generalized Bunce–Deddens algebras
HTML articles powered by AMS MathViewer

by Stefanos Orfanos PDF

Proc. Amer. Math. Soc. 138 (2010), 299-308 Request permission

Abstract:

We define a broad class of crossed product C*-algebras of the form $C(\tilde {G})\rtimes G$, where $G$ is a discrete countable amenable residually finite group, and $\tilde {G}$ is a profinite completion of $G$. We show that they are unital separable simple nuclear quasidiagonal C*-algebras, of real rank zero, stable rank one, with comparability of projections and with a unique trace.

References

Lawrence G. Brown and Gert K. Pedersen, $C^*$-algebras of real rank zero, J. Funct. Anal. 99 (1991), no. 1, 131–149. MR 1120918, DOI 10.1016/0022-1236(91)90056-B
Nathanial P. Brown and Narutaka Ozawa, $C^*$-algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, RI, 2008. MR 2391387, DOI 10.1090/gsm/088
Christopher Deninger and Klaus Schmidt, Expansive algebraic actions of discrete residually finite amenable groups and their entropy, Ergodic Theory Dynam. Systems 27 (2007), no. 3, 769–786. MR 2322178, DOI 10.1017/S0143385706000939
Thierry Giordano, Ian Putnam, and Christian Skau, Affable equivalence relations and orbit structure of Cantor dynamical systems, Ergodic Theory Dynam. Systems 24 (2004), no. 2, 441–475. MR 2054051, DOI 10.1017/S014338570300066X
David W. Kribs, Inductive limit algebras from periodic weighted shifts on Fock space, New York J. Math. 8 (2002), 145–159. MR 1934386
Orfanos, S., Quasidiagonality of crossed products, Journal of Operator Theory, to appear.
N. Christopher Phillips, Crossed products of the Cantor set by free minimal actions of $\Bbb Z^d$, Comm. Math. Phys. 256 (2005), no. 1, 1–42. MR 2134336, DOI 10.1007/s00220-004-1171-y
Jean-Paul Pier, Amenable locally compact groups, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 767264
Jean Renault, A groupoid approach to $C^{\ast }$-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. MR 584266
Marc A. Rieffel, Dimension and stable rank in the $K$-theory of $C^{\ast }$-algebras, Proc. London Math. Soc. (3) 46 (1983), no. 2, 301–333. MR 693043, DOI 10.1112/plms/s3-46.2.301
Benjamin Weiss, Monotileable amenable groups, Topology, ergodic theory, real algebraic geometry, Amer. Math. Soc. Transl. Ser. 2, vol. 202, Amer. Math. Soc., Providence, RI, 2001, pp. 257–262. MR 1819193, DOI 10.1090/trans2/202/18
Dana P. Williams, Crossed products of $C{^\ast }$-algebras, Mathematical Surveys and Monographs, vol. 134, American Mathematical Society, Providence, RI, 2007. MR 2288954, DOI 10.1090/surv/134
John S. Wilson, Profinite groups, London Mathematical Society Monographs. New Series, vol. 19, The Clarendon Press, Oxford University Press, New York, 1998. MR 1691054