Property (SP) and Inclusions of C*-algebras

Abstract

As a generalization of Gabe and Neagu's inclusions of real rank zero, we introduce a notion of Property (SP) for inclusions of C*-algebras. We begin the systematic study of SP-inclusions, establishing a topological description in the commutative setting and permanence under many constructions. We also extend known results about the permanence of Property (SP) for C*-algebras under classical symmetries to the setting of quantum symmetries; we show that Property (SP) for (simple, separable) C*-algebras is preserved under irreducible C*-discrete inclusions. Finally, we resolve (in the negative) a question raised by Gabe and Neagu about inclusions coming from Furstenburg boundaries. Namely, we show the existence of a countable, discrete, non-amenable group $G$ such that $C_r^*(G) \subseteq C(\partial_F G) \rtimes_r G$ is not an SP-inclusion (and therefore not an inclusion of real rank zero).