C^*-Algebra relations - Loring, Terry A. (MATHEMATICA SCANDINAVICA, 2010)
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.