Nowhere scattered C*-algebras

TL;DR

This paper introduces the concept of nowhere scattered C*-algebras, characterizes it through multiple perspectives, and explores its implications under certain rank conditions, revealing new divisibility properties.

Contribution

It provides a comprehensive characterization of nowhere scattered C*-algebras and links this property to spectral, divisibility, and structural features, including rank conditions.

Findings

Characterization of nowhere scattered C*-algebras via spectrum and Cuntz semigroup

Connection between nowhere scatteredness and absence of elementary quotients

Stronger divisibility properties under real rank zero or stable rank one

Abstract

We say that a C*-algebra is nowhere scattered if none of its quotients contains a minimal open 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.