标题： 一个绝对局部紧致到的卷积代数 显示英文标题

标题： The convolution algebra of an absolutely locally compact topos

摘要： 我们引入了一类称为“绝对局部紧”的到域以及在其上的“可接受”层环类。 对于任何这样的环化到域 $(\mathcal{T},A)$ ，我们附加了一个对合卷积代数 $\mathcal{C}_c(\mathcal{T},A)$ ，它在Morita等价的意义下是良好定义的，并且其特征是 $\mathcal{C}_c(\mathcal{T},A)$ 上非退化模的范畴与 $A$-模层在 $\mathcal{T}$上的范畴等价。 在$A$是实数或复数 Dedekind 数的层的情况下，我们在这个对合代数上构造了几个范数，使其可以完成为各种 Banach 代数和$C^*$-代数： $L^1(\mathcal{T},A)$，$C^*_{red}(\mathcal{T},A)$和 $C^*_{max}(\mathcal{T},A)$。 我们还给出了一些例子，说明这种构造对应于已知的对合代数构造，如群胚卷积代数和 Leavitt 路径代数。 显示英文摘要

摘要： 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 $C^*$-algebras: $L^1(\mathcal{T},A)$, $C^*_{red}(\mathcal{T},A)$ and $C^*_{max}(\mathcal{T},A)$. We also give some examples where this construction corresponds to well known constructions of involutive algebras, like groupoids convolution algebra and Leavitt path algebras.