An introduction to the Global Glimm Problem

TL;DR

This survey introduces the Global Glimm Problem in C*-algebra theory, highlighting its significance and recent progress, and discusses its implications for broader open questions in the field.

Contribution

It provides a concise overview of the problem, key concepts, methods, and the potential impact of its complete resolution on related open questions.

Findings

Summarizes partial solutions and key concepts related to the Global Glimm Problem.

Highlights the importance of resolving the problem for understanding regularity properties of C*-algebras.

Discusses the implications for dimension reduction phenomena and conjectures in the field.

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.