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Communications in Mathematical Physics

, Volume 331, Issue 1, pp 215–238 | Cite as

Characterizations of Categories of Commutative C*-Subalgebras

  • Chris Heunen
Article

Abstract

We aim to characterize the category of injective *-homomorphisms between commutative C*-subalgebras of a given C*-algebra A. We reduce this problem to finding a weakly terminal commutative subalgebra of A, and solve the latter for various C*-algebras, including all commutative ones and all type I von Neumann algebras. This addresses a natural generalization of the Mackey–Piron programme: which lattices are those of closed subspaces of Hilbert space? We also discuss the way this categorified generalization differs from the original question.

Keywords

Inverse Semigroup Closed Subspace Orthomodular Lattice Compact Hausdorff Space Commutative Subalgebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Blass A., Sagan B.E.: Möbius functions of lattices. Adv. Math. 127, 94–123 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bunge M.: Internal presheaves toposes. Cahiers de topologie et géométrie différentielle catégoriques 18(3), 291–330 (1977)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Bures, D.: Abelian Subalgebras of Von Neumann Algebras. Number 110 in Memoirs. American Mathematical Society, Providence (1971)Google Scholar
  4. 4.
    Davidson K.R.: C*-algebras by example. American Mathematical Society, Providence (1991)Google Scholar
  5. 5.
    Döring, A., Isham, C.J.: Topos methods in the foundations of physics. In: Halvorson, H. (ed.) Deep Beauty. Cambridge University Press, Cambridge (2011)Google Scholar
  6. 6.
    Firby P.A.: Lattices and compactifications I. Proc. Lond. Math. Soc. 27, 22–50 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Firby P.A.: Lattices and compactifications II. Proc. Lond. Math. Soc. 27, 51–60 (1973)CrossRefzbMATHGoogle Scholar
  8. 8.
    Funk J.: Semigroups and toposes. Semigr. Forum 75, 480–519 (2007)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Hamhalter Jan.: Isomorphisms of ordered structures of abelian C*-subalgebras of C*-algebras. J. Math. Anal. Appl. 383, 391–399 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Hamhalter J., Turilova E.: Structure of associative subalgebras of Jordan operator algebras. Q. J. Math. 64(2), 397–408 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Harding, J., Döring, A.: Abelian subalgebras and the Jordan structure of a von Neumann algebra. Houst. J. Math. (2014, in press)Google Scholar
  12. 12.
    Heunen C.: Complementarity in categorical quantum mechanics. Found. Phys. 42(7), 856–873 (2012)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Heunen C., Landsman N.P., Spitters B.: A topos for algebraic quantum theory. Commun. Math. Phys. 291, 63–110 (2009)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Heunen, C., Landsman, N.P., Spitters, B.: Bohrification. In: Halvorson, H. (ed.) Deep Beauty. Cambridge University Press, Cambridge (2011)Google Scholar
  15. 15.
    Heunen C., Reyes M.L.: Diagonalizing matrices over AW*-algebras. J. Funct. Anal. 264(8), 1873–1898 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Heunen C., Reyes M.L.: Active lattices determine AW*-algebras. J. Math. Anal. Appl. 416, 289–313 (2014)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Johnstone P.T.: Sketches of an Elephant: A Topos Theory Compendium. Oxford University Press, Oxford (2002)Google Scholar
  18. 18.
    Kadison R.V., Ringrose J.R.: Fundamentals of the Theory of Operator Algebras. Academic Press, London (1983)zbMATHGoogle Scholar
  19. 19.
    Kalmbach G.: Orthomodular Lattices. Academic Press, London (1963)Google Scholar
  20. 20.
    Kalmbach G.: Measures and Hilbert lattices. World Scientific, Singapore (1986)CrossRefzbMATHGoogle Scholar
  21. 21.
    Mac Lane S., Moerdijk I.: Sheaves in Geometry and Logic. Springer, Berlin (1992)CrossRefzbMATHGoogle Scholar
  22. 22.
    Piron, C.: Foundations of Quantum Physics. Number 19 in Mathematical Physics Monographs. W.A. Benjamin, Reading (1976)Google Scholar
  23. 23.
    Rédei M.: Quantum Logic in Algebraic Approach. Kluwer, Dordrecht (1998)CrossRefzbMATHGoogle Scholar
  24. 24.
    Segal, I.: Decompositions of Operator Algebras II: Multiplicity Theory. Number 9 in Memoirs. American Mathematical Society, Providence (1951)Google Scholar
  25. 25.
    Sinclair, A.M., Smith, R.R.: Finite von Neumann Algebras and Masas. Number 351 in London Mathematical Society lecture notes. Cambridge University Press, Cambridge (2008)Google Scholar
  26. 26.
    Solèr M.P.: Characterization of Hilbert spaces by orthomodular spaces. Commun. Algebra 23(1), 219–243 (1995)CrossRefzbMATHGoogle Scholar
  27. 27.
    Tomiyama J.: On some types of maximal abelian subalgebras. J. Funct. Anal. 10, 373–386 (1972)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    van den Berg B., Heunen C.: No-go theorems for functorial localic spectra of noncommutative rings. Electron. Proc. Theor. Comput. Sci. 95, 21–25 (2012)CrossRefGoogle Scholar
  29. 29.
    van den Berg B., Heunen C.: Noncommutativity as a colimit. Appl. Categ. Struct. 20(4), 393–414 (2012)CrossRefzbMATHGoogle Scholar
  30. 30.
    Wilce, A.: Test spaces. In: Handbook of Quantum Logic, vol. II. Elsevier, Amsterdam (2008)Google Scholar
  31. 31.
    Yoon Y.-J.: Characterizations of partition lattices. Bull. Korean Math. Soc. 31(2), 237–242 (1994)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.University of OxfordOxfordUK

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