The universal property of infinite direct sums in C${}^{\ast }$-categories and W${}^{\ast }$-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 C${}^{\ast }$-categories, provided that one uses enrichment in Banach spaces. We then formulate such a universal property for infinite direct sums in C${}^{\ast }$-categories, and prove the equivalence with the existing definition due to Ghez, Lima and Roberts in the case of W${}^{\ast }$-categories. These infinite direct sums specialize to the usual ones in the category of Hilbert spaces, and more generally in any W${}^{\ast }$-category of normal representations of a W${}^{\ast }$-algebra. Finding a universal property for the more general case of direct integrals remains an open problem.

11 pages