C*-Algebra Relations

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 σ-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.