Deformation quantization for actions of Qpd

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 [4], aims to extend Rieffel's deformation theory [27] for more general groups than Rd. In [4], 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 [12] and further extended by Bechata [1] and Unterberger [34].