Soft operators in C*-algebras

Journal article, 2024

We say that a C⁎-algebra is soft if it has no nonzero unital quotients, and we connect this property to the Hjelmborg-Rørdam condition for stability and to property (S) of Ortega-Perera-Rørdam. We further say that an operator in a C⁎-algebra is soft if its associated hereditary subalgebra is, and we provide useful spectral characterizations of this concept. Of particular interest are C⁎-algebras that have an abundance of soft elements in the sense that every hereditary subalgebra contains an almost full soft element. We show that this property is implied by the Global Glimm Property, and that every C⁎-algebra with an abundance of soft elements is nowhere scattered. This sheds new light on the long-standing Global Glimm Problem of whether every nowhere scattered C⁎-algebra has the Global Glimm Property.