A separable Brown-Douglas-Fillmore theorem and weak stability

Author:
Huaxin Lin

Translated by:

Journal:
Trans. Amer. Math. Soc. **356** (2004), 2889-2925

MSC (2000):
Primary 46L05, 46L80

DOI:
https://doi.org/10.1090/S0002-9947-04-03558-5

Published electronically:
March 2, 2004

MathSciNet review:
2052601

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Abstract: We give a separable Brown-Douglas-Fillmore theorem. Let be a separable amenable -algebra which satisfies the approximate UCT, be a unital separable amenable purely infinite simple -algebra and be two monomorphisms. We show that and are approximately unitarily equivalent if and only if We prove that, for any and any finite subset , there exist and a finite subset satisfying the following: for any amenable purely infinite simple -algebra and for any contractive positive linear map such that

for all there exists a homomorphism such that

provided, in addition, that are finitely generated. We also show that every separable amenable simple -algebra with finitely generated -theory which is in the so-called bootstrap class is weakly stable with respect to the class of amenable purely infinite simple -algebras. As an application, related to perturbations in the rotation -algebras studied by U. Haagerup and M. Rørdam, we show that for any irrational number and any there is such that in any unital amenable purely infinite simple -algebra if

for a pair of unitaries, then there exists a pair of unitaries and in such that

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Additional Information

**Huaxin Lin**

Affiliation:
Department of Mathematics, East China Normal University, Shanghai, People’s Republic of China

Address at time of publication:
Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222

DOI:
https://doi.org/10.1090/S0002-9947-04-03558-5

Keywords:
Weakly semiprojective $C^*$-algebras,
purely infinite simple $C^*$-algebras

Received by editor(s):
September 18, 2002

Received by editor(s) in revised form:
April 29, 2003

Published electronically:
March 2, 2004

Additional Notes:
This research was partially supported by NSF grant DMS 0097903

Article copyright:
© Copyright 2004
American Mathematical Society