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

  • Tobias FritzEmail author
  • Bas Westerbaan


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.


Direct sum Biproduct \(\hbox {C}^*\)-category \(\hbox {W}^*\)-category Category of Hilbert spaces 

Mathematics Subject Classification

Primary: 46M15 Secondary: 18E05 46L10 


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We thank Robert Furber for copious help and discussion, as well as Chris Heunen and Martti Karvonen for discussion. Part of this work was conducted while the first author was at the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  2. 2.Department of Computer ScienceUniversity College LondonLondonUK

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