Local and global proper infiniteness for continuous C(X)-algebras

10 pagesAll 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 that for all integer $n\geq 3$, there is a state $\psi_n:\mathcal{T}_n\to\C$ such that the reduced unital free product $(\mathcal{T}_n, \psi_n)\ast_\C(\mathcal{T}_n, \psi_n)$ is a K$_1$-injective \cst-algebra which contains the algebraic free product $\mathcal{T}_n\circledast_\C\mathcal{T}_n\,$