Deformation Quantization for actions of $\mathbb{Q}_p^{d}$

The main objective of this article is to develop the theory of deformation of $C^*$-algebras endowed with a group action, from the perspective of non-formal equivariant quantization. This program, initiated in \cite{Bieliavsky-Gayral}, aims to extend Rieffel's deformation theory \cite{Ri} for more general groups than $\mathbb R^d$. In \cite{Bieliavsky-Gayral}, we have constructed such a theory for a class of non-Abelian Lie groups. In the present article, we study the somehow opposite situation of Abelian but non-Lie groups. More specifically, we construct here a deformation theory of $C^*$-algebras endowed with an action of a finite dimensional vector space over a non-Archimedean local field of characteristic different from 2. At the root of our construction stands the $p$-adic version of the Weyl quantization introduced by Haran and further extended by Bechata and Unterberger.