(Almost) C*-algebras as sheaves with self-action
Cecilia Flori and Tobias Fritz
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Submission date: 03. Dec. 2015 (revised version: October 2017)
published in: Journal of noncommutative geometry, 11 (2017) 3, p. 1069-1113
MSC-Numbers: 46L05, 46L60, 18F20, 20A05
Keywords and phrases: Axiomatics of C*-algebras, sheaf theory, algebraic quantum mechanics, topos quantum theory
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Via Gelfand duality, a unital C*-algebra A induces a functor from compact Hausdorﬀ spaces to sets, CHaus → Set. We show how this functor encodes standard functional calculus in A as well as its multivariate generalization. Certain sheaf conditions satisﬁed by this functor provide a further generalization of functional calculus. Considering such sheaves CHaus → Set abstractly, we prove that the piecewise C*-algebras of van den Berg and Heunen are equivalent to a full subcategory of the category of sheaves, where a simple additional constraint characterizes the objects in the subcategory. It is open whether this additional constraint holds automatically, in which case piecewise C*-algebras would be the same as sheaves CHaus → Set.
Intuitively, these structures capture the commutative aspects of C*-algebra theory. In order to ﬁnd a complete reaxiomatization of unital C*-algebras within this language, we introduce almost C*-algebras as piecewise C*-algebras equipped with a notion of inner automorphisms in terms of a self-action. We provide some evidence for the conjecture that the forgetful functor from unital C*-algebras to almost C*-algebras is fully faithful, and ask whether it is an equivalence of categories. We also develop an analogous notion of almost group, and prove that the forgetful functor from groups to almost groups is not full.
In terms of quantum physics, our work can be seen as an attempt at a reconstruction of quantum theory from physically meaningful axioms, as realized by Hardy and others in a diﬀerent framework. Our ideas are inspired by and also provide new input for the topos-theoretic approach to quantum theory.