Infinite norm decompositions of C$^*$-algebras

TL;DR

This paper explores the concept of infinite norm decomposition in C$^*$-algebras, demonstrating that such decompositions yield new C$^*$-algebras and constructing examples of factors and purely infinite algebras.

Contribution

It introduces the notion of infinite norm decomposition for C$^*$-algebras and proves that these decompositions form new C$^*$-algebras, expanding the understanding of their structure.

Findings

Infinite norm decomposition of any C$^*$-algebra is itself a C$^*$-algebra.

Constructed examples of C$^*$-factors with infinite and nonzero finite projections.

Constructed simple purely infinite C$^*$-algebras.

Abstract

In the given article the notion of infinite norm decomposition of a C$^*$-algebra is investigated. The norm decomposition is some generalization of Peirce decomposition. It is proved that the infinite norm decomposition of any C$^*$-algebra is a C$^*$-algebra. C$^*$-factors with an infinite and a nonzero finite projection and simple purely infinite C$^*$-algebras are constructed.