Categorical criterion for existence of universal $C^*$--algebras

Abstract

We deal with categories, which determine universal $C^*$--algebras. These categories are called the compact $C^*$--relations. They were introduced by T.A.~Loring. Given a set $X,$ a compact $C^*$--relation on $X$ is a category, the objects of which are functions from $X$ to $C^*$--algebras, and morphisms are $\ast$--homomorphisms of $C^*$--algebras making the appropriate triangle diagrams commute. Moreover, these functions and $\ast$--homo\-mor\-phisms satisfy certain axioms. In this article, we prove that every compact $C^*$--relation is both complete and cocomplete. As an appli\-cation of the completeness of compact $C^*$--relations, we obtain the criterion for the existence of universal $C^*$--algebras.