(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 \emph{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.