Matrix stability and Morita invariance

Let $G$ be a group.

We prove that matrix stability for either $G$-algebras or $G$-graded algebras guarantees Morita invariance.

As a consequence, bivariant algebraic K-theory (either $G$-equivariant or $G$-graded) is Morita invariant.

In particular, we show that if $G$ is a finite group acting freely on a finite simplicial set $X$, then $\ell^X\rtimes G$ and $\ell^{X/G}$ are kk-equivalent.

Here, $\ell^Y$ denotes the $\ell$-algebra of piecewise polynomial functions on $Y$ with coefficients in the ground ring $\ell$.