Weak Semiprojectivity in Purely Infinite Simple C*-Algebras

Lin, Huaxin

Canadian journal of mathematics, Tome 59 (2007) no. 2, pp. 343-371

Résumé

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 $a\,\in \,\mathcal{F}$ .

Détail

DOI : 10.4153/CJM-2007-015-9

Mots-clés : 46L05, 46L80, weakly semiprojective, purely infinite simple, C*-algebras

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