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We introduce a class of toposes called "absolutely locally compact" toposesand 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 equivalenceand 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 orcomplex Dedekind numbers, we construct several norms on this involutive algebrathat allows to complete it in various Banach and $C^*$-algebras:$L^1(\mathcal{T},A)$, $C^*_{red}(\mathcal{T},A)$ and$C^*_{max}(\mathcal{T},A)$. We also give some examples where this constructioncorresponds to well known constructions of involutive algebras, like groupoidsconvolution algebra and Leavitt path algebras.

The convolution algebra of an absolutely locally compact topos