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Residually finite actions and crossed products

Published online by Cambridge University Press:  17 October 2011

DAVID KERR and
PIOTR W. NOWAK
DAVID KERR
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA (email: kerr@math.tamu.edu, pnowak@math.tamu.edu)
PIOTR W. NOWAK
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA (email: kerr@math.tamu.edu, pnowak@math.tamu.edu)
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Abstract

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We study a notion of residual finiteness for continuous actions of discrete groups on compact Hausdorff spaces and how it relates to the existence of norm microstates for the reduced crossed product. Our main result asserts that an action of a free group on a zero-dimensional compact metrizable space is residually finite if and only if its reduced crossed product admits norm microstates, i.e., is an MF algebra.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 5 , October 2012, pp. 1585 - 1614
Copyright
Copyright © Cambridge University Press 2011

References

[1]Anantharaman-Delaroche, C. and Renault, J.. Amenable Groupoids (Monographies de L’Enseignement Mathématique, 36). L’Enseignement Mathématique, Geneva, 2000.Google Scholar
[2]Blackadar, B.. Operator Algebras. Theory of C *-Algebras and von Neumann Algebras (Encyclopaedia of Mathematical Sciences, 122). Springer, Berlin, 2006, Operator Algebras and Non-commutative Geometry, III.Google Scholar
[3]Blackadar, B. and Kirchberg, E.. Irreducible representations of inner quasidiagonal C *-algebras. arXiv:0711.4949.Google Scholar
[4]Blackadar, B. and Kirchberg, E.. Inner quasidiagonality and strong NF algebras. Pacific J. Math. 198 (2001), 307–329.CrossRefGoogle Scholar
[5]Blackadar, B. and Kirchberg, E.. Generalized inductive limits and quasidiagonality. C *-algebras (Münster, 1999). Springer, Berlin, 2000, pp. 23–41.CrossRefGoogle Scholar
[6]Blackadar, B. and Kirchberg, E.. Generalized inductive limits of finite-dimensional C *-algebras. Math. Ann. 307 (1997), 343–380.CrossRefGoogle Scholar
[7]Bowen, L.. The ergodic theory of free group actions: entropy and the f-invariant. Groups Geom. Dyn. 4 (2010), 419–432.CrossRefGoogle Scholar
[8]Bowen, L.. Measure conjugacy invariants for actions of countable sofic groups. J. Amer. Math. Soc. 23 (2010), 217–245.CrossRefGoogle Scholar
[9]Brown, N. P.. On quasidiagonal C *-algebras. Operator Algebras and Applications (Advanced Studies in Pure Mathematics, 38). Mathematical Society Japan, Tokyo, 2004, pp. 19–64.CrossRefGoogle Scholar
[10]Brown, N. P., Dykema, K. J. and Jung, K.. Free entropy dimension in amalgamated free products. Proc. Lond. Math. Soc. 97 (2008), 339–367.CrossRefGoogle Scholar
[11]Brown, N. P. and Ozawa, N.. C *-Algebras and Finite-Dimensional Approximations. (Graduate Studies in Mathematics, 88). American Mathematical Society, Providence, RI, 2008.CrossRefGoogle Scholar
[12]Conley, C.. Isolated Invariant Sets and the Morse Index (CBMS Regional Conference Series in Mathematics, 38). American Mathematical Society, Providence, RI, 1978.CrossRefGoogle Scholar
[13]Cuntz, J. and Pedersen, G. K.. Equivalence and traces on C *-algebras. J. Funct. Anal. 33 (1979), 135–164.CrossRefGoogle Scholar
[14]Dadarlat, M.. On the approximation of quasidiagonal C *-algebras. J. Funct. Anal. 167 (1999), 69–78.CrossRefGoogle Scholar
[15]Davidson, K. R.. C *-algebras by Example (Fields Institute Monographs, 6). American Mathematical Society, Providence, RI, 1996.CrossRefGoogle Scholar
[16]Eckhardt, C.. On Oℒ∞ structure of nuclear, quasidiagonal C *-algebras. J. Funct. Anal. 258 (2010), 1–19.CrossRefGoogle Scholar
[17]Gong, G. and Lin, H.. Almost multiplicative morphisms and almost commuting matrices. J. Operator Theory 40 (1998), 217–275.Google Scholar
[18]Gootman, E. C.. Primitive ideals of C *-algebras associated with transformation groups. Trans. Amer. Math. Soc. 170 (1972), 97–108.Google Scholar
[19]Gootman, E. C. and Rosenberg, J.. The structure of crossed product C *-algebras: a proof of the generalized Effros–Hahn conjecture. Invent. Math. 52 (1979), 283–298.CrossRefGoogle Scholar
[20]Haagerup, U. and Thorbjørnsen, S.. A new application of random matrices: Ext(C *red(F 2)) is not a group. Ann. of Math. (2) 162 (2005), 711–775.CrossRefGoogle Scholar
[21]Hadwin, D.. Strongly quasidiagonal C *-algebras. J. Operator Theory 18 (1987), 3–18, With an appendix by J. Rosenberg.Google Scholar
[22]Junge, M., Ozawa, N. and Ruan, Z.-J.. On Oℒ∞ structures of nuclear C *-algebras. Math. Ann. 325 (2003), 449–483.CrossRefGoogle Scholar
[23]Kerr, D. and Li, H.. Soficity, amenability, and dynamical entropy. Amer. J. Math. to appear.Google Scholar
[24]Kerr, D. and Li, H.. Entropy and the variational principle for actions of sofic groups. Invent. Math. to appear.Google Scholar
[25]Lin, H.. AF-embeddings of the crossed products of AH-algebras by finitely generated abelian groups. Int. Math. Res. Pap. IMRP 2008 (2008), Art. ID rpn007, 67 pages.Google Scholar
[26]Lubotzky, A. and Shalom, Y.. Finite representations in the unitary dual and Ramanujan groups. Discrete Geometric Analysis (Contemporary Mathematics, 347). American Mathematical Society, Providence, RI, 2004, pp. 173–189.CrossRefGoogle Scholar
[27]Margulis, G.. Free subgroups of the homeomorphism group of the circle. C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), 669–674.CrossRefGoogle Scholar
[28]Margulis, G. and Vinberg, E. B.. Some linear groups virtually having a free quotient. J. Lie Theory 10 (2000), 171–180.Google Scholar
[29]Orfanos, S.. Quasidiagonality of crossed products. J. Operator Theory 66 (2011), 209–216.Google Scholar
[30]Pimsner, M. V.. Embedding some transformation group C *-algebras into AF-algebras. Ergod. Th. & Dynam. Sys. 3 (1983), 613–626.CrossRefGoogle Scholar
[31]Pimsner, M. and Voiculescu, D.. Exact sequences for K-groups and Ext-groups of certain cross-product C *-algebras. J. Operator Theory 4 (1980), 93–118.Google Scholar
[32]Rosenblatt, J. M.. Invariant measures and growth conditions. Trans. Amer. Math. Soc. 193 (1974), 33–53.CrossRefGoogle Scholar
[33]Rørdam, M. and Sierakowski, A.. Purely infinite C *-algebras arising from crossed products. Ergod. Th. & Dynam. Sys. to appear.Google Scholar
[34]Salinas, N.. Homotopy invariance of Ext(A). Duke Math. J. 44 (1977), 777–794.CrossRefGoogle Scholar
[35]Sauvageot, J.-L.. Idéaux primitifs de certains produits croisés. Math. Ann. 231 (1977), 61–76.CrossRefGoogle Scholar
[36]Tomiyama, J.. The Interplay Between Topological Dynamics and Theory of C *-Algebras (Lecture Notes Series, 2). Res. Inst. Math., Seoul, 1992.Google Scholar
[37]Voiculescu, D.. A non-commutative Weyl–von Neumann theorem. Rev. Roum. Math. Pures Appl. 21 (1976), 97–113.Google Scholar
[38]Voiculescu, D.. A note on quasidiagonal C *-algebras and homotopy. Duke Math. J. 62 (1991), 267–271.CrossRefGoogle Scholar
[39]Wagon, S.. The Banach–Tarski Paradox. Cambridge University Press, Cambridge, 1993.Google Scholar
[40]Williams, D. P.. Crossed Products of C *-Algebras (Mathematical Surveys and Monographs, 134). American Mathematical Society, Providence, RI, 2007.CrossRefGoogle Scholar
[41]Zeller-Meier, G.. Produits croisés d’une C *-algèbre par un groupe d’automorphismes. J. Math. Pures Appl. (9) 49 (1968), 101–239.Google Scholar