The convolution algebra of an absolutely locally compact topos

Abstract

We introduce a class of toposes called "absolutely locally compact" toposes and of "admissible" sheaf of rings over such toposes. To any such ringed topos $(\mathcal{T},A)$ we attach an involutive convolution algebra ${\mathcal{C}}_c(\mathcal{T},A)$ which is well defined up to Morita equivalence and characterized by the fact that the category of non-degenerate modules over ${\mathcal{C}}_c(\mathcal{T},A)$ is equivalent to the category of sheaf of $A$-module over $\mathcal{T}$. In the case where $A$ is the sheaf of real or complex Dedekind numbers, we construct several norms on this involutive algebra that allows to complete it in various Banach and $C^{\ast }$-algebras: $L^1(\mathcal{T},A)$, $C_{red}^{\ast }(\mathcal{T},A)$ and $C_{max}^{\ast }(\mathcal{T},A)$. We also give some examples where this construction corresponds to well known constructions of involutive algebras, like groupoids convolution algebra and Leavitt path algebras.

36 pages

References in corpus (2)

• Stack semantics and the comparison of material and structural set theories
• Constructive Gelfand duality for non-unital commutative C*-algebras