On properties of compact $C^\ast$-relations

The talk is devoted to the categories introduced by T.A. Loring in the framework of an axiomatic approach to universal $C^\ast$-algebras. These categories are called $C^\ast$-relations. For a given set $X$, a $C^\ast$-relation on $X$ is a category whose objects are functions from $X$ to $C^\ast$-algebras and morphisms are $\ast$-homomorphisms of $C^\ast$-algebras making the appropriate triangle diagrams commute. Moreover, the functions and the $\ast$-homomorphisms satisfy certain axioms. Those $C^*$-relations that have initial objects are said to be compact. The universal $C^*$-algebra for a compact $C^\ast$-relation is defined as its initial object. To study properties of compact $C^\ast$-relations, we construct functors between these categories. Among the $C^\ast$-relations, we consider $\ast$-polynomial relations associated with $\ast$-polynomial pairs. It is shown that every $C^*$-algebra is a universal $C^*$-algebra defined by a $\ast$-polynomial pair. Using the above-mentioned functors, we prove that every compact $C^*$-relation is isomorphic to a $\ast$-polynomial relation. Further, it is shown that every compact $C^*$-relation is both complete and cocomplete. As an application of the completeness of compact $C^*$-relations, we obtain a criterion for the existence of universal $C^*$-algebras.
The talk is based on the results of our joint work with I.S. Berdnikov, E.V. Lipacheva and K.A. Shishkin.