On the classification problem for C*-algebras

TL;DR

This paper introduces new classification notions for C*-algebras, establishing their relation to von Neumann types, and provides a comprehensive classification theorem for C*-factors based on these types.

Contribution

It develops a new classification framework for C*-algebras, linking them to von Neumann algebra types and proving key decomposition and characterization theorems.

Findings

GCR-algebras are of von Neumann type I

NGCR-algebras lack nonzero Abelian annihilators

C*-factors of type I$_n$ are classified as such

Abstract

In the given article it is introduced new notions of a C$^*$-algebra of von Neumann type I and C$^*$-algebras of types I$_n$, II, II$_1$, II$_\infty$ and III. It is proved that any GCR-algebra is a C$^*$-algebra of von Neumann type I, and a C$^*$-algebra is an NGCR-algebra if and only if this C$^*$-algebra does not have a nonzero Abelian annihilator. Also an analog of the theorem on decomposition of a von Neumann algebra to subalgebras of types I, II and III is proved. In the final part it is proved that every C$^*$-factor of von Neumann type I is a C$^*$-algebra of type I$_n$ for some cardinal number $n$, every simple C$^*$-algebra of type II$_1$ is finite, every simple purely infinite C$^*$-algebra is of type III and every W$^*$-factor of type II$_\infty$ has a simple C$^*$-subalgebra of type II$_\infty$. Finally it is formulated a classification theorem for C$^*$-factors.