$C^*$-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 $\sigma$-$C^{*}$-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.