Equivariant $KK$-theory of Bernoulli shifts on $C^*$-algebras with approximately inner flip

Abstract: Building on Enders--Schemeitat--Tikuisis' classification, we show that a separable $C^*$-algebra $A$ with approximately inner flip in the UCT class is $K$-theoretically self-absorbing if and only if for every finite group $G$, the Bernoulli shift on $A^{\otimes G}$ is $KK^G$-equivalent to the trivial action. This in particular applies to UHF-algebras of infinite type and computes the $K$-theory of the associated crossed product. Along the way, we obtain an alternative proof of Hirshberg--Winter's result that the Bernoulli shift of $G$ on a UHF-algebra of infinite type absorbs the trivial action up to conjugacy. For more general amenable groups $G$, we develop $K$-theory formulas for Bernoulli shifts on UHF-absorbing $C^*$-algebras, and establish $KK^G$-triviality for Bernoulli shifts on strongly self-absorbing $C^*$-algebras satisfying the UCT.