Andreas Doering: "Some steps towards noncommutative Gel'fand duality"

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• published: 01 May 2012
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Speaker: Andreas Doering (University of Oxford) Title: Some steps towards noncommutative Gel'fand duality Event: Quantum Physics and Logic 2010 (May 2010, University of Oxford) Abstract: Gel'fand duality is one of the central mathematical insights of the last century. Each abelian C*-algebra A gives rise to a compact or locally compact Hausdorff space, the Gel'fand spectrum Sigma^A. Conversely, each compact or locally compact Hausdorff space X determines an abelian C*-algebra C(X) of continuous functions. In quantum theory, noncommutative C*-algebras play a central role, but we are still lacking a good notion of spectrum for these algebras. Such a spectrum would be a suitable noncommutative space and would provide quantum theory with a geometrical underpinning that is absent so far. In previous work, it was shown that the spectral presheaf Sigma^A associated with an arbitrary unital C*- or von Neumann algebra A has many properties of a spectrum. Here we show that the assignment of Sigma^A to A is functorial in a suitable sense and can be seen as the first half of a noncommutative version of Gel'fand duality. We show that for abelian algebras, our construction reduces to ordinary Gel'fand duality. Moreover, it is shown how the group of unitary operators in a von Neumann algebra is faithfully represented by automorphisms of the (set of subobjects of the) spectral presheaf.