Geodesic
Categorical criterion for existence of universal $C^*$--algebras
Ufa mathematical journal, Tome 16 (2024) no. 3, pp. 113-124

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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 $\ast$–homomorphisms satisfy certain axioms. In this article, we prove that every compact $C^*$–relation is both complete and cocomplete. As an application of the completeness of compact $C^*$–relations, we obtain the criterion for the existence of universal $C^*$–algebras.
Keywords: compact $C^*$–relation, complete category, universal $C^*$–algebra. 
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     title = {Categorical criterion for existence  of universal $C^*$--algebras},
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R. N. Gumerov; E. V. Lipacheva; K. A. Shishkin. Categorical criterion for existence  of universal $C^*$--algebras. Ufa mathematical journal, Tome 16 (2024) no. 3, pp. 113-124. http://geodesic.mathdoc.fr/item/UFA_2024_16_3_a9/