Relative commutants of strongly self-absorbing C∗-algebras

The relative commutant A′∩AU$$A^{\prime }\cap A^{\mathcal U}$$ of a strongly self-absorbing algebra A is indistinguishable from its ultrapower AU$$A^{\mathcal U}$$. This applies both to the case when A is the hyperfinite II1$$_1$$ factor and to the case when it is a strongly self-absorbing C∗$$\mathrm {C}^*$$-algebra. In the latter case, we prove analogous results for ℓ∞(A)/c0(A)$$\ell _\infty (A)/c_0(A)$$ and reduced powers corresponding to other filters on N$${\mathbb N}$$. 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.