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Weak Semiprojectivity in Purely Infinite Simple C*-Algebras

Published online by Cambridge University Press:  20 November 2018

Huaxin Lin
Huaxin Lin*
Affiliation:
Department of Mathematics, East China Normal University, Shangai, P.R. China, and Department of Mathematics, University of Oregon, Eugene, OR 97403-1222, U.S.A. email: hlin@uoregon.edu
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Abstract

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Let $A$ be a separable amenable purely infinite simple ${{C}^{*}}$-algebra which satisfies the Universal Coefficient Theorem. We prove that $A$ is weakly semiprojective if and only if ${{K}_{i}}(A\text{)}$ is a countable direct sum of finitely generated groups $\left( i\,=\,0,\,1 \right)$. Therefore, if $A$ is such a ${{C}^{*}}$-algebra, for any $\varepsilon \,>\,0$ and any finite subset $\mathcal{F}\,\subset \,A$ there exist $\delta \,>\,0$ and a finite subset $G\,\subset \,A$ satisfying the following: for any contractive positive linear map $L\,:\,A\,\to \,B$ (for any ${{C}^{*}}$-algebra $B$) with $||L\left( ab \right)\,-\,L\left( a \right)L\left( b \right)||\,<\,\delta$ for $a,\,b\,\in \,\mathcal{G}$ there exists a homomorphism $h:\,A\,\to \,B$ such that $||\,h\left( a \right)\,-\,L\left( a \right)||\,<\,\varepsilon$ for $a\,\in \,\mathcal{F}$.

Keywords

46L0546L80weakly semiprojectivepurely infinite simpleC*-algebras
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 2 , 01 April 2007 , pp. 343 - 371
Copyright
Copyright © Canadian Mathematical Society 2007

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