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.

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Acknowledgements

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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Correspondence to Tobias Fritz.

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Fritz, T., Westerbaan, B. The Universal Property of Infinite Direct Sums in \(\hbox {C}^*\)-Categories and \(\hbox {W}^*\)-Categories. Appl Categor Struct 28, 355–365 (2020). https://doi.org/10.1007/s10485-019-09583-9

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Keywords

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

Mathematics Subject Classification

  • Primary: 46M15
  • Secondary: 18E05
  • 46L10