Local and global proper infiniteness for continuous C(X)-algebras
Résumé
All unital continuous \cst-bundles with properly infinite fibres are properly infinite \cst-algebras if and only if the full unital free product $\Td\ast_\C\Td$ of two copies of the Cuntz extensions $\Td$ generated by two isometries with orthogonal ranges is a K$_1$-injective \cst-algebra (\cite[Theorem 5.5]{BRR08}, \cite[Proposition 4.2]{Blan10}). We show in this article that there is a state $\psi_n$ with faithful GNS representation on the Cuntz extension $\mathcal{T}_n$ such that the reduced unital free product $(\mathcal{T}_n, \psi_n)\ast_\C(\mathcal{T}_n, \psi_n)$ is K$_1$-injective for all $n\geq 3$.
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