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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: https://doi.org/10.1090/S0065-9266-09-00611-5
Published electronically: December 14, 2009
MSC: Primary 46L05, 46L35

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Table of Contents

Chapters

• 1. Prelude
• 2. The Basic Homotopy Lemma for higher dimensional spaces
• 3. Purely infinite simple $C^*$-algebras
• 4. Approximate homotopy
• 5. Super Homotopy
• 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 $\operatorname {Bott}(h,u)=0,$ 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 \,\,\, \mathrm {and}\,\,\, \|[h(g),u_t]\|<\epsilon \,\,\,\, \mathrm {for}\,\,\mathrm {all}\,\,\, g\in {\mathcal F}\,\,\,\mathrm {and}\,\,\, t\in [0,1]. \end{equation} Moreover, \begin{eqnarray} \mathrm {Length}(\{u_t\})\le 2\pi +\epsilon . \end{eqnarray} We show that if $\operatorname {dim}X\le 1,$ or $A$ is purely infinite simple, then $\delta$ and ${\mathcal G}$ are universal (independent of $A$ or $h$). In the case that $\operatorname {dim} X=1,$ 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 $\operatorname {dim} X\ge 2.$ 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.