Topological entropy, embeddings and unitaries in nuclear quasidiagonal $C^*$-algebras
HTML articles powered by AMS MathViewer
- by Nathanial P. Brown PDF
- Proc. Amer. Math. Soc. 128 (2000), 2603-2609 Request permission
Abstract:
Using topological entropy of automorphisms of $C^*$-algebras, it is shown that some important facts from the theory of AF algebras do not carry over to the class of $A\mathbb {T}$ algebras. It is shown that in general one cannot perturb a basic building block into a larger one which almost contains it. The same entropy obstruction used to prove this fact also provides a new obstruction to the known fact that two injective homomorphisms from a building block into an $A\mathbb {T}$ algebra need not differ by an (inner) automorphism when they agree on K-theory.References
- Bruce Blackadar and Eberhard Kirchberg, Generalized inductive limits of finite-dimensional $C^*$-algebras, Math. Ann. 307 (1997), no. 3, 343–380. MR 1437044, DOI 10.1007/s002080050039
- B. Blackadar and E. Kirchberg, Inner quasidiagonality and strong NF algebras, Preprint.
- Ola Bratteli, Erling Størmer, Akitaka Kishimoto, and Mikael Rørdam, The crossed product of a UHF algebra by a shift, Ergodic Theory Dynam. Systems 13 (1993), no. 4, 615–626. MR 1257025, DOI 10.1017/S0143385700007574
- N.P. Brown, AF embeddability of crossed products of AF algebras by the integers, J. Funct. Anal. 160 (1998), 150–175.
- N.P. Brown, Topological Entropy in Exact $C^*$-algebras, Preprint 1998.
- Kenneth R. Davidson, $C^*$-algebras by example, Fields Institute Monographs, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1402012, DOI 10.1090/fim/006
- George A. Elliott, On the classification of $C^*$-algebras of real rank zero, J. Reine Angew. Math. 443 (1993), 179–219. MR 1241132, DOI 10.1515/crll.1993.443.179
- George A. Elliott and David E. Evans, The structure of the irrational rotation $C^*$-algebra, Ann. of Math. (2) 138 (1993), no. 3, 477–501. MR 1247990, DOI 10.2307/2946553
- Akitaka Kishimoto, The Rohlin property for automorphisms of UHF algebras, J. Reine Angew. Math. 465 (1995), 183–196. MR 1344136, DOI 10.1515/crll.1995.465.183
- A. Kishimoto, Unbounded derivations in $A\mathbb { T}$ algebras, J. Funct. Anal. 160 (1998), 270–311.
- Q. Lin and N.C. Phillips, Ordered K-theory for $C^*$-algebras of minimal homeomorphisms, Advances in Operator Algebras and Operator Theory, Contem. Math. 228 (1998), 289–314.
- Mihai V. Pimsner, Embedding some transformation group $C^{\ast }$-algebras into AF-algebras, Ergodic Theory Dynam. Systems 3 (1983), no. 4, 613–626. MR 753927, DOI 10.1017/S0143385700002182
- 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
- Ian F. Putnam, On the topological stable rank of certain transformation group $C^*$-algebras, Ergodic Theory Dynam. Systems 10 (1990), no. 1, 197–207. MR 1053808, DOI 10.1017/S0143385700005484
- Dan Voiculescu, Almost inductive limit automorphisms and embeddings into AF-algebras, Ergodic Theory Dynam. Systems 6 (1986), no. 3, 475–484. MR 863206, DOI 10.1017/S0143385700003618
- Dan Voiculescu, Dynamical approximation entropies and topological entropy in operator algebras, Comm. Math. Phys. 170 (1995), no. 2, 249–281. MR 1334396
Additional Information
- Nathanial P. Brown
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1901
- Address at time of publication: Department of Mathematics, University of California-Berkeley, Berkeley, California 94720
- Email: nbrown@math.purdue.edu
- Received by editor(s): October 14, 1998
- Published electronically: March 1, 2000
- Additional Notes: This work was partially supported by an NSF Dissertation Enhancement Award
- Communicated by: David R. Larson
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 2603-2609
- MSC (1991): Primary 46L05, 46L80, 46L55
- DOI: https://doi.org/10.1090/S0002-9939-00-05329-6
- MathSciNet review: 1664305