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Oxford Quantum Talks Archive

Some steps towards noncommutative Gel'fand duality

Andreas Doering, University of Oxford
Quantum Physics and Logic 2010, May 2010, University of Oxford

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.

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