Relative commutants of strongly self-absorbing C*-algebras

The relative commutant $A^{\mathrm{\prime }}\cap A^{\mathcal{U}}$ 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 ${\mathrm{\ell }}_{\mathrm{\infty }}(A)\mathrm{/}{c}_0(A)$ and reduced powers corresponding to other filters on $\mathbf{N}$. 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.