The Universal Property of Infinite Direct Sums in $\hbox {C}^*$-Categories and $\hbox {W}^*$-Categories

Abstract

When formulating universal properties for objects in a dagger category, one usually expects a universal property to characterize the universal object up to unique unitary isomorphism. We observe that this is automatically the case in the important special case of $\hbox {C}^*$-categories, provided that one uses enrichment in Banach spaces. We then formulate such a universal property for infinite direct sums in $\hbox {C}^*$-categories, and prove the equivalence with the existing definition due to Ghez, Lima and Roberts in the case of $\hbox {W}^*$-categories. These infinite direct sums specialize to the usual ones in the category of Hilbert spaces, and more generally in any $\hbox {W}^*$-category of normal representations of a $\hbox {W}^*$-algebra. Finding a universal property for the more general case of direct integrals remains an open problem.