Relative commutants of strongly self-absorbing C∗-algebras

The relative commutant A ′ ∩A U A′∩AU of a strongly self-absorbing algebra A is indistinguishable from its ultrapower A U AU. This applies both to the case when A is the hyperfinite II 1 1 factor and to the case when it is a strongly self-absorbing C ∗ C∗-algebra. In the latter case, we prove analogous results for ℓ ∞ (A)/c 0 (A) ℓ∞(A)/c0(A) and reduced powers corresponding to other filters on N 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.