The Gelfand spectrum of a noncommutative \(C^*\)-algebra: a topos-theoretic approach. (English) Zbl 1223.46062

Heunen, Chris; Landsman, Nicolaas P.; Spitters, Bas; Wolters, Sander

J. Aust. Math. Soc. 90, No. 1, 39-52 (2011).

The celebrated Gelfand duality says that compact Hausdorff spaces make an opposite category to the unital commutative \(C^*\)-algebras; if the \(C^*\)-algebras are no longer commutative, then the duality extends to the so-called ‘generalized spaces’. The generalized space \(X:=\text{Spec}~A\) can hardly be a conventional topological space, but thanks to A. Grothendieck it can be successfully treated in terms of the sheaves over \(X\); such sheaves (of sets) are vernacularly known as topoi. The aim of the paper under review is to establish an analog of the Gelfand duality for the generalized space \(X\) using the technique of topoi; the foundation of such an approach has been laid by C. Heunen, N. P. Landsman and B. Spitters in [“A topos for algebraic quantum theory”, Commun. Math. Phys. 291, No. 1, 63–110 (2009; Zbl 1209.81147)].
The authors take a noncommutative \(C^*\)-algebra \(A\) and look at the partially ordered set \({\mathcal C}(A)\) of unital commutative \(C^*\)-sub-algebras of \(A\) endowed with the Alexandrov topology; the topos \(Sh~({\mathcal C}(A))\) is considered. The main results are the duality theorems for the so-called internal and external Gelfand spectra of \(A\) given in terms of \(Sh~({\mathcal C}(A))\) and also a bijective correspondence between points of \(\text{Spec}~A\) and valuations on \(A\); as an example, \(A=M_n({\mathbb C})\) is studied. Overall, the paper makes a good reading, being very instructive and sprinkled with interesting (philosophical) observations.

Reviewer: Igor V. Nikolaev (Ottawa)

MSC:

46L85

Noncommutative topology

Keywords:

Gelfand duality; category; topoi

Citations:

Zbl 1209.81147

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