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Weak Semiprojectivity in Purely Infinite Simple C*-Algebras
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- Journal:
- Canadian Journal of Mathematics / Volume 59 / Issue 2 / 01 April 2007
- Published online by Cambridge University Press:
- 20 November 2018, pp. 343-371
- Print publication:
- 01 April 2007
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- Article
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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 
$a\,\in \,\mathcal{F}$.
