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


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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  1. 1.

    Borceux, F.: Handbook of Categorical Algebra. 2. Categories and Structures. Encyclopedia of Mathematics and its Applications, vol. 51. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  2. 2.

    Gel’fand, I.M., Vilenkin, N.Y.: Generalized Functions: Applications of Harmonic Analysis, vol. 4. Academic Press, New York (1964). (Translated by Amiel Feinstein)

    Google Scholar 

  3. 3.

    Ghez, P., Lima, R., Roberts, J.E.: \(\text{ W }^*\)-categories. Pac. J. Math. 120, 79–109 (1985)

    Article  Google Scholar 

  4. 4.

    Heunen, C., Karvonen, M.: Limits in dagger categories. Theory Appl. Categ. 34(18), 468–513 (2019). arXiv:1803.06651

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras, vol. 2. American Mathematical Society, Providence (1997)

    Google Scholar 

  6. 6.

    Kelly, G.M.: Basic Concepts of Enriched Category Theory. Reprints in Theory and Applications of Categories, vol. 10. Cambridge University Press, Cambridge (2005). (Reprint of the 1982 original)

    Google Scholar 

  7. 7.

    Paschke, W.L.: Inner product modules over b*-algebras. Trans. Am. Math. Soc. 182, 443–468 (1973)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Rieffel, M.A.: Morita equivalence for C*-algebras and W*-algebras. Journal of Pure and Applied Algebra 5(1), 51–96 (1974)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Sakai, S.: \(C^*\)-Algebras and \(W^*\)-Algebras. Classics in Mathematics. Springer, Berlin (1998). Reprint of the 1971 edition

    Google Scholar 

  10. 10.

    Westerbaan, A.: The Category of Von Neumann Algebras. PhD thesis, Radboud University (2019). arXiv:1804.02203

  11. 11.

    Westerbaan, A., Westerbaan, B.: A universal property for sequential measurement. J Math Phys 57(9), 092203 (2016)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Westerbaan, B.: Dagger and dilations in the category of von Neumann algebras. PhD thesis, Radboud University (2019). arXiv:1803.01911

Download references


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).

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

Mathematics Subject Classification

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