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The Universal Property of Infinite Direct Sums in C∗ -Categories and W∗ -Categories

Fritz, T; Westerbaan, B; (2020) The Universal Property of Infinite Direct Sums in C∗ -Categories and W∗ -Categories. Applied Categorical Structures , 28 pp. 355-365. 10.1007/s10485-019-09583-9. Green open access

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

Type: Article
Title: The Universal Property of Infinite Direct Sums in C∗ -Categories and W∗ -Categories
Open access status: An open access version is available from UCL Discovery
DOI: 10.1007/s10485-019-09583-9
Publisher version: http://dx.doi.org/10.1007/s10485-019-09583-9
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: Direct sum, Biproduct, C∗ -category, W∗ -category, Category of Hilbert spaces
UCL classification: UCL
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URI: https://discovery.ucl.ac.uk/id/eprint/10084765
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