Non-amenable tight squeezes by Kirchberg algebras (Q2134199)

An inclusion \(A \subseteq B\) of \(C^*\)-algebras \(A\) and \(B\) is said to be rigid if the identity map on \(B\) is the only completely positive map on \(B\) which is the identity on \(A\). In a main theorem, the author proves that given any simple, unital, separable and purely infinite \(C^*\)-algebra \((A,\alpha,{\mathbf F}_\infty)\) equipped with an approximately inner action \(\alpha\) by the free group \({\mathbf F}_\infty\), there exists a perturbation \(\alpha'\) of \(\alpha\) by an inner action such that for any simple \(C^*\)-algebra \((B,\beta,{\mathbf F}_\infty)\) with nonzero fixed point algebra it holds that the canonical inclusion \[ B \rtimes_{r,\beta} {\mathbf F}_\infty \subseteq (A \otimes_{\min} B) \rtimes_{r,\alpha' \otimes \beta} {\mathbf F}_\infty \] is rigid and no other \(C^*\)-subalgebra is properly between them. Several related theorems follow. For example, a Kirchberg algebra \(B\) can be squeezed into \(C^*\)-algebras \(A \subseteq B \subseteq C\) without proper intermediate \(C^*\)-algebras such that \(A\) is simple, purely infinite and non-nuclear, and \(C\) is simple, purely infinite and non-exact, and the inclusions are rigid and actually \(KK\)-equivalences. Moreover, the reduced crossed product \(A \rtimes_{r,\alpha} {\mathbf F}_\infty\) admits a \(KK\)-equivalent, rigid embedding into a Kirchberg algebra without intermediate \(C^*\)-algebra if \(A\) is simple, separable and nuclear with nonzero fixed point algebra.