C*-Algebra Relations

TL;DR

This paper introduces and axiomatizes C*-relations, including compact and closed types, linking them to universal C*-algebras and zero-sets, with applications to lifting problems.

Contribution

It defines C*-relations axiomatically, introduces the concept of closed C*-relations, and connects these to zero-sets in free sigma-C*-algebras, advancing the theory of relations in C*-algebras.

Findings

Closed C*-relations correspond to zero-sets in free sigma-C*-algebras

Compact C*-relations determine universal C*-algebras

Applications to lifting problems are discussed

Abstract

We investigate relations on elements in C*-algebras, including *-polynomial relations, order relations and all relations that correspond to universal C*-algebras. We call these C*-relations and define them axiomatically. Within these are the compact C*-relations, which are those that determine universal C*-algebras, and we introduce the more flexible concept of a closed C*-relation. In the case of a finite set of generators, we show that closed C*-relations correspond to the zero-sets of elements in a free sigma -C*-algebra. This provides a solid link between two of the previous theories on relations in C*-algebras. Applications to lifting problems are briefly considered in the last section.