We prove that a separable, nuclear, purely infinite, simple $C^*$-algebra satisfying the universal coefficient theorem is weakly semiprojective if and only if its $K$-groups are direct sums of cyclic groups.

Keywords:

Kirchberg algebra, weak semiprojectivity, graph $C^*$-algebra Kirchberg algebra, weak semiprojectivity, graph $C^*$-algebra

MSC Classifications:

46L05, 46L80, 22A22 show english descriptions General theory of $C^*$-algebras
$K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
Topological groupoids (including differentiable and Lie groupoids) [See also 58H05] 46L05 - General theory of $C^*$-algebras
46L80 - $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
22A22 - Topological groupoids (including differentiable and Lie groupoids) [See also 58H05]

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