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Approximate homotopy of homomorphisms from $C(X)$ into a simple $C^*$-algebra


About this Title

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

Publication: Memoirs of the American Mathematical Society
Publication Year 2010: 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 $C^*$-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 $X$ be a finite CW complex and let $h_1, h_2: C(X)\to A$ be two unital homomorphisms, where $A$ is a unital $C^*$-algebra. We study the problem when $h_1$ and $h_2$ are approximately homotopic. We present a $K$-theoretical necessary and sufficient condition for them to be approximately homotopic under the assumption that $A$ is a unital separable simple $C^*$-algebra, of tracial rank zero, or $A$ is a unital purely infinite simple $C^*$-algebra. When they are approximately homotopic, we also give an upper bound for the length of the homotopy. Suppose that $h: C(X)\to A$ is a monomorphism and $u\in A$ is a unitary (with $[u]=0$ in $K_1(A)$). We prove that, for any $\epsilon >0,$ and any compact subset ${\mathcal F}\subset C(X),$ there exist $\delta >0$ and a finite subset ${\mathcal G}\subset C(X)$ satisfying the following: if $\|[h(f), u]\|<\delta $ for all $f\in {\mathcal G}$ and $\text {Bott}(h,u)=0,$ then there exists a continuous rectifiable path $\{u_t: t\in [0,1]\}$ in $A$ such that \begin{equation} u_{0}=u, u_{1}=1_{A} and |[h(g),u_{t}]|<𝜀 for all g∈\mathcal F and t∈[0,1]. \end{equation} Moreover, \begin{eqnarray} Length({u_{t}})\le2𝜋+𝜀. \end{eqnarray} We show that if $\text {dim}X\le 1,$ or $A$ is purely infinite simple, then $\delta $ and ${\mathcal G}$ are universal (independent of $A$ or $h$). In the case that ${\rm dim} X=1,$ this provides an improvement of the so-called Basic Homotopy Lemma of Bratteli, Elliott, Evans and Kishimoto for the case that $A$ is as mentioned above. Moreover, we show that $\delta $ and ${\mathcal G}$ cannot be universal whenever $\text {dim} X\ge 2.$ Nevertheless, we also found that $\delta $ can be chosen to be dependent on a measure distribution but independent of $A$ and $h.$ The above version of the so-called Basic Homotopy is also extended to the case that $C(X)$ is replaced by an AH-algebra. We also present some general versions of the so-called Super Homotopy Lemma.

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