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Approximate homotopy of homomorphisms from into a simple -algebra


About this Title

Huaxin Lin, Department of Mathematics, East China Normal University, Shanghai, China

Publication: Memoirs of the American Mathematical Society
Publication Year 2009: Volume 205, Number 963
ISBNs: 978-0-8218-5194-4 (print); 978-1-4704-0577-9 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-09-00611-5
Published electronically: December 14, 2009
MathSciNet review: 2643313
MSC (2000): Primary 46L05; Secondary 46L35, 46L80

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Table of Contents


Chapters

  • Chapter 1. Prelude
  • Chapter 2. The Basic Homotopy Lemma for higher dimensional spaces
  • Chapter 3. Purely infinite simple -algebras
  • Chapter 4. Approximate homotopy
  • Chapter 5. Super homotopy
  • Chapter 6. Postlude

Abstract


In this paper we prove Generalized Homotopy Lemmas. These type of results play an important role in the classification theory of -homomorphisms up to asymptotic unitary equivalence.

Let be a finite CW complex and let be two unital homomorphisms, where is a unital -algebra. We study the problem when and are approximately homotopic. We present a -theoretical necessary and sufficient condition for them to be approximately homotopic under the assumption that is a unital separable simple -algebra, of tracial rank zero, or is a unital purely infinite simple -algebra. When they are approximately homotopic, we also give an upper bound for the length of the homotopy.

Suppose that is a monomorphism and is a unitary (with in ). We prove that, for any and any compact subset there exist and a finite subset satisfying the following: if for all and then there exists a continuous rectifiable path in such that

Moreover,

We show that if or is purely infinite simple, then and are universal (independent of or ). In the case that this provides an improvement of the so-called Basic Homotopy Lemma of Bratteli, Elliott, Evans and Kishimoto for the case that is as mentioned above. Moreover, we show that and cannot be universal whenever Nevertheless, we also found that can be chosen to be dependent on a measure distribution but independent of and The above version of the so-called Basic Homotopy is also extended to the case that is replaced by an AH-algebra.

We also present some general versions of the so-called Super Homotopy Lemma.

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