Relative commutants of strongly self-absorbing C*-algebras

TL;DR

This paper demonstrates that the relative commutant of strongly self-absorbing algebras is essentially the same as their ultrapower, with implications for classification and structure of such algebras.

Contribution

It establishes that the relative commutant of strongly self-absorbing C*-algebras matches their ultrapower, extending known results and providing new classification insights.

Findings

Relative commutant equals ultrapower for strongly self-absorbing algebras.

Results apply to hyperfinite II$_1$ factor and C*-algebras.

Strongly self-absorbing algebras are smoothly classifiable.

Abstract

The relative commutant $A'\cap A^{\mathcal{U}}$ of a strongly self-absorbing algebra $A$ is indistinguishable from its ultrapower $A^{\mathcal{U}}$. This applies both to the case when $A$ is the hyperfinite II$_1$ factor and to the case when it is a strongly self-absorbing C*-algebra. In the latter case we prove analogous results for $\ell_\infty(A)/c_0(A)$ and reduced powers corresponding to other filters on $\bf N$. Examples of algebras with approximately inner flip and approximately inner half-flip are provided, showing the optimality of our results. We also prove that strongly self-absorbing algebras are smoothly classifiable, unlike the algebras with approximately inner half-flip.