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Residually finite actions and crossed products

  • DAVID KERR (a1) and PIOTR W. NOWAK (a1)


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.



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[1]Anantharaman-Delaroche, C. and Renault, J.. Amenable Groupoids (Monographies de L’Enseignement Mathématique, 36). L’Enseignement Mathématique, Geneva, 2000.
[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.
[3]Blackadar, B. and Kirchberg, E.. Irreducible representations of inner quasidiagonal C *-algebras. arXiv:0711.4949.
[4]Blackadar, B. and Kirchberg, E.. Inner quasidiagonality and strong NF algebras. Pacific J. Math. 198 (2001), 307329.
[5]Blackadar, B. and Kirchberg, E.. Generalized inductive limits and quasidiagonality. C *-algebras (Münster, 1999). Springer, Berlin, 2000, pp. 2341.
[6]Blackadar, B. and Kirchberg, E.. Generalized inductive limits of finite-dimensional C *-algebras. Math. Ann. 307 (1997), 343380.
[7]Bowen, L.. The ergodic theory of free group actions: entropy and the f-invariant. Groups Geom. Dyn. 4 (2010), 419432.
[8]Bowen, L.. Measure conjugacy invariants for actions of countable sofic groups. J. Amer. Math. Soc. 23 (2010), 217245.
[9]Brown, N. P.. On quasidiagonal C *-algebras. Operator Algebras and Applications (Advanced Studies in Pure Mathematics, 38). Mathematical Society Japan, Tokyo, 2004, pp. 1964.
[10]Brown, N. P., Dykema, K. J. and Jung, K.. Free entropy dimension in amalgamated free products. Proc. Lond. Math. Soc. 97 (2008), 339367.
[11]Brown, N. P. and Ozawa, N.. C *-Algebras and Finite-Dimensional Approximations. (Graduate Studies in Mathematics, 88). American Mathematical Society, Providence, RI, 2008.
[12]Conley, C.. Isolated Invariant Sets and the Morse Index (CBMS Regional Conference Series in Mathematics, 38). American Mathematical Society, Providence, RI, 1978.
[13]Cuntz, J. and Pedersen, G. K.. Equivalence and traces on C *-algebras. J. Funct. Anal. 33 (1979), 135164.
[14]Dadarlat, M.. On the approximation of quasidiagonal C *-algebras. J. Funct. Anal. 167 (1999), 6978.
[15]Davidson, K. R.. C *-algebras by Example (Fields Institute Monographs, 6). American Mathematical Society, Providence, RI, 1996.
[16]Eckhardt, C.. On O structure of nuclear, quasidiagonal C *-algebras. J. Funct. Anal. 258 (2010), 119.
[17]Gong, G. and Lin, H.. Almost multiplicative morphisms and almost commuting matrices. J. Operator Theory 40 (1998), 217275.
[18]Gootman, E. C.. Primitive ideals of C *-algebras associated with transformation groups. Trans. Amer. Math. Soc. 170 (1972), 97108.
[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), 283298.
[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), 711775.
[21]Hadwin, D.. Strongly quasidiagonal C *-algebras. J. Operator Theory 18 (1987), 318, With an appendix by J. Rosenberg.
[22]Junge, M., Ozawa, N. and Ruan, Z.-J.. On O structures of nuclear C *-algebras. Math. Ann. 325 (2003), 449483.
[23]Kerr, D. and Li, H.. Soficity, amenability, and dynamical entropy. Amer. J. Math. to appear.
[24]Kerr, D. and Li, H.. Entropy and the variational principle for actions of sofic groups. Invent. Math. to appear.
[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.
[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. 173189.
[27]Margulis, G.. Free subgroups of the homeomorphism group of the circle. C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), 669674.
[28]Margulis, G. and Vinberg, E. B.. Some linear groups virtually having a free quotient. J. Lie Theory 10 (2000), 171180.
[29]Orfanos, S.. Quasidiagonality of crossed products. J. Operator Theory 66 (2011), 209216.
[30]Pimsner, M. V.. Embedding some transformation group C *-algebras into AF-algebras. Ergod. Th. & Dynam. Sys. 3 (1983), 613626.
[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), 93118.
[32]Rosenblatt, J. M.. Invariant measures and growth conditions. Trans. Amer. Math. Soc. 193 (1974), 3353.
[33]Rørdam, M. and Sierakowski, A.. Purely infinite C *-algebras arising from crossed products. Ergod. Th. & Dynam. Sys. to appear.
[34]Salinas, N.. Homotopy invariance of Ext(A). Duke Math. J. 44 (1977), 777794.
[35]Sauvageot, J.-L.. Idéaux primitifs de certains produits croisés. Math. Ann. 231 (1977), 6176.
[36]Tomiyama, J.. The Interplay Between Topological Dynamics and Theory of C *-Algebras (Lecture Notes Series, 2). Res. Inst. Math., Seoul, 1992.
[37]Voiculescu, D.. A non-commutative Weyl–von Neumann theorem. Rev. Roum. Math. Pures Appl. 21 (1976), 97113.
[38]Voiculescu, D.. A note on quasidiagonal C *-algebras and homotopy. Duke Math. J. 62 (1991), 267271.
[39]Wagon, S.. The Banach–Tarski Paradox. Cambridge University Press, Cambridge, 1993.
[40]Williams, D. P.. Crossed Products of C *-Algebras (Mathematical Surveys and Monographs, 134). American Mathematical Society, Providence, RI, 2007.
[41]Zeller-Meier, G.. Produits croisés d’une C *-algèbre par un groupe d’automorphismes. J. Math. Pures Appl. (9) 49 (1968), 101239.

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Residually finite actions and crossed products

  • DAVID KERR (a1) and PIOTR W. NOWAK (a1)


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