Nowhere scattered C*-algebras

Abstract

We say that a C * -algebra is nowhere scattered if none of its quotients contains a minimal projection. We characterize this property in various ways, by topological properties of the spectrum, by divisibility properties in the Cuntz semigroup, by the existence of Haar unitaries for states, and by the absence of nonzero ideal-quotients that are elementary, scattered or type I.Under the additional assumption of real rank zero or stable rank one, we show that nowhere scatteredness implies even stronger divisibility properties of the Cuntz semigroup.

Cuntz semigroups of ultraproduct C∗‐algebras

“…A C*-algebra is said to be nowhere scattered if it has no nonzero, elementary ideal-quotients; see [TV21b].…”

“…The category Cu was introduced in [CEI08] and was extensively studied in [APT18, APP18, APT20a, APT20b, APT20c, APRT22] as well as [TV22a,TV21a,TV21b]. The cones of functionals on Cu-semigroups have also been thoroughly studied; see, for example, [ERS11,Rob13,APRT21].…”

The Global Glimm Property

“…We show that a C * -algebra has the Global Glimm Property if and only if its Cuntz semigroup is (2, ω)-divisible; see Theorem 3.6. On the other hand, by [TV21b,Theorem 8.9], a C * -algebra is nowhere scattered if and only if its Cuntz semigroup is weakly (2, ω)-divisible; see Paragraph 2.4. Thus, one can reformulate the Global Glimm Problem as: Is every weakly (2, ω)-divisible Cuntz semigroup (2, ω)-divisible?…”