(Almost) C*-algebras as sheaves with self-action

Abstract

Via Gelfand duality, a unital C*-algebra $A$ induces a functor from compact Hausdorff spaces to sets, $\mathsf{CHaus}\to\mathsf{Set}$. 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 $\mathsf{CHaus}\to\mathsf{Set}$ 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 $\mathsf{CHaus}\to\mathsf{Set}$. 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.