Structure and classification of $C^{\ast }$-algebras

Abstract

We give an overview of the development over the last 15 years of the theory of simple C*-algebras, in particular in regards to their classification and structure. We discuss dimension theory for (simple) C*-algebras, in particular the so-called stable and real ranks, and we explain how properties of C*-algebras of low dimension (stable rank one and real rank zero) was used by the author and P. Friis to give a new and simple proof of a theorem of H. Lin that almost commuting self-adjoint matrices are close to exactly commuting self-adjoint matrices. Elliott’s classification program is explained and is contrasted with recent examples of C*-algebras of “high dimension”, including an example of a simple C*-algebras with a finite and an infinite projection.