Nowhere scattered C*-algebras

Scattered topological spaces and their C*-analogs, known as scattered
C*-algebras, have been studied since the 70's and admit a number of
interesting characterizations. In this talk, I will define nowhere
scattered C*-algebras as, informally, those C*-algebras that are very
far from being scattered. I will then characterize this property in
various ways, such as the absence of nonzero elementary ideal-quotients,
topological properties of the spectrum, and divisibility properties in
the Cuntz semigroup. Further, I will also show that these divisibility
properties can be strengthened in the real rank zero or the stable rank
one case.

The talk is based on joint work with Hannes Thiel.