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