Eduard Vilalta (Universitat Autònoma de Barcelona)
Nowhere scattered multiplier algebras
Abstract: A natural assumption that ensures sufficient noncommutativity of a C*-algebra is nowhere scatteredness, which in one of its many formulations asks the algebra to contain no nonzero elementary ideal-quotients. This notion enjoys many good permanence properties, but fails to pass to certain unitizations. For example, no minimal unitization of a non-unital C*-algebra (nowhere scattered or not) can ever be nowhere scattered. However, it is unclear when a nowhere scattered C*-algebra has a nowhere scattered multiplier algebra.
In this talk, I will give sufficient conditions under which this happens. It will follow from the main result of the talk that a $\sigma$-unital C*-algebra of finite nuclear dimension, or of real rank zero, or of stable rank one and k-comparison, is nowhere scattered if and only if its multiplier algebra is. I will also give some examples of nowhere scattered C*-algebras whose multiplier algebra is not nowhere scattered.