Topological entropy, embeddings and unitaries in nuclear quasidiagonal -algebras

Author:
Nathanial P. Brown

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

Published electronically:
March 1, 2000

MathSciNet review:
1664305

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Using topological entropy of automorphisms of -algebras, it is shown that some important facts from the theory of AF algebras do not carry over to the class of 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 algebra need not differ by an (inner) automorphism when they agree on K-theory.

**[BK1]**B. Blackadar and E. Kirchberg,*Generalized inductive limits of finite dimensional C*-algebras*, Math. Ann.**307**(1997), 343 - 380. MR**98c:46112****[BK2]**B. Blackadar and E. Kirchberg,*Inner quasidiagonality and strong NF algebras*, Preprint.**[BKRS]**O. Bratteli, A. Kishimoto, M. Rrdam and E. Strmer,*The crossed product of a UHF algebra by a shift*, Ergod. Th. Dynam. Sys.**13**(1993), 615 - 626. MR**95c:46111****[Br1]**N.P. Brown,*AF embeddability of crossed products of AF algebras by the integers*, J. Funct. Anal.**160**(1998), 150-175. CMP**99:06****[Br2]**N.P. Brown,*Topological Entropy in Exact**-algebras*, Preprint 1998.**[Da]**K.R. Davidson,*-algebras by Example*, Fields Inst. Monographs vol. 6, Amer. Math. Soc., (1996). MR**97i:46095****[El]**G.A. Elliott,*On the classification of**-algebras of real rank zero*, J. reine angew. Math.**443**(1993), 179-219. MR**94i:46074****[EE]**G.A. Elliott and D.E. Evans,*The structure of the irrational rotation**-algebra*, Ann. Math.**138**(1993), 477-501. MR**94j:46066****[Ki1]**A. Kishimoto,*The Rohlin property for automorphisms of UHF algebras*, J. reine angew. Math.**465**(1995), 183 - 196. MR**96k:46114****[Ki2]**A. Kishimoto,*Unbounded derivations in**algebras*, J. Funct. Anal.**160**(1998), 270-311. CMP**99:06****[LP]**Q. Lin and N.C. Phillips,*Ordered K-theory for**-algebras of minimal homeomorphisms*, Advances in Operator Algebras and Operator Theory, Contem. Math.**228**(1998), 289-314.**[Pi]**M. Pimsner,*Embedding some transformation group C**-algebras into AF algebras*, Ergod. Th. Dynam. Sys.**3**(1983), 613 - 626. MR**86d:46054****[PV]**M. Pimsner and D. Voiculescu,*Exact sequences for K-groups and Ext-groups of certain crossed products of*, J. Oper. Th.**4**(1980), 93 - 118. MR**82c:46074****[Pu]**I. Putnam,*On the topological stable rank of certain transformation group C*-algebras*, Ergod. Th. and Dyn. Sys.**10**(1990), 197 - 207. MR**91f:46090****[Vo1]**D. Voiculescu,*Almost inductive limit automorphisms and embeddings into AF algebras*, Ergod. Th. Dynam. Sys.**6**(1986), 475 - 484. MR**88k:46073****[Vo2]**D. Voiculescu,*Dynamical approximation entropies and topological entropy in operator algebras*, Comm. Math. Phys.**170**(1995), 249 - 281. MR**97b:46082**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
46L05,
46L80,
46L55

Retrieve articles in all journals with MSC (1991): 46L05, 46L80, 46L55

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

DOI:
https://doi.org/10.1090/S0002-9939-00-05329-6

Keywords:
Topological entropy,
(generalized) inductive limits,
inner automorphisms,
embeddings

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

Article copyright:
© Copyright 2000
American Mathematical Society