An introduction to the Global Glimm Problem

Abstract

The Global Glimm Problem lies at the heart of several open questions regarding regularity properties of C*-algebras. The problem has been open for over two decades, and has recently garnered significant attention due to its strong ties to non-simple versions of well-known dimension reduction phenomena for simple C*-algebras, such as the weakly purely infinite problem and the non-simple Toms-Winter conjecture. This survey offers a concise introduction to the problem, detailing some of the key concepts and methods used in the partial solutions achieved thus far. Particular emphasis is placed on the implications that a complete resolution of the Global Glimm Problem would have for other open questions in the field.