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Homomorphisms from $C(X)$ into $C^*$-algebras

  Published:1997-10-01
 Printed: Oct 1997
  • Huaxin Lin
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Abstract

Let $A$ be a simple $C^*$-algebra with real rank zero, stable rank one and weakly unperforated $K_0(A)$ of countable rank. We show that a monomorphism $\phi\colon C(S^2) \to A$ can be approximated pointwise by homomorphisms from $C(S^2)$ into $A$ with finite dimensional range if and only if certain index vanishes. In particular, we show that every homomorphism $\phi$ from $C(S^2)$ into a UHF-algebra can be approximated pointwise by homomorphisms from $C(S^2)$ into the UHF-algebra with finite dimensional range. As an application, we show that if $A$ is a simple $C^*$-algebra of real rank zero and is an inductive limit of matrices over $C(S^2)$ then $A$ is an AF-algebra. Similar results for tori are also obtained. Classification of ${\bf Hom}\bigl(C(X),A\bigr)$ for lower dimensional spaces is also studied.
Keywords: Homomorphism of $C(S^2)$, approximation, real, rank zero, classification Homomorphism of $C(S^2)$, approximation, real, rank zero, classification
MSC Classifications: 46L05, 46L80, 46L35 show english descriptions General theory of $C^*$-algebras
$K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
Classifications of $C^*$-algebras
46L05 - General theory of $C^*$-algebras
46L80 - $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
46L35 - Classifications of $C^*$-algebras
 

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