Selfless reduced free product $C^{\ast }$-algebras

We study selflessness in the general setting of reduced free products of $C^{\ast }$-algebras. Towards this end, we develop a suitable theory of rapid decay for filtrations in arbitrary $C^{\ast }$-probability spaces. We provide several natural examples and permanence properties of this phenomenon. By using this framework in combination with von Neumann algebraic techniques involving approximate forms of orthogonality, we are able to prove selflessness for general families of reduced free product $C^{\ast }$-algebras. As an instance of our results, we prove selflessness and thus strict comparison for the canonical $C^{\ast }$-algebras generated by Voiculescu’s free semicircular systems. Our results also provide new examples of purely infinite reduced free products.