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Characterizations of categories of commutative C*-subalgebras

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Piron’s problem in the foundations of quantum mechanics asks which lattices are those of closed subspaces of Hilbert space. We solve a natural categorification, by characterizing the category of injective -homomorphisms between commutative C-subalgebras of various C-algebras, namely C-algebras of operators on separable Hilbert spaces, any finite-dimensional C-algebra, and any commutative C-algebra. This is an important step in a promising ongoing approach to a noncommutative generalisation of Gelfand’s duality between compact Hausdorff spaces and commutative C*-algebras, that we will outline.

This talk is part of the Category Theory Seminar series.

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