ON p-ADIC OPERATOR ALGEBRAS

Building on the recent definition of a p-adic analogue of a separable Hilbert space by Thom and Claussnitzer, we introduce non-separable p-adic Hilbert spaces and define an algebra of bounded operators on such spaces. This sets up the study of involutive Banach algebras that can be isometrically represented on p-adic Hilbert spaces, which we call p-adic operator algebras. We then show that several known examples of C*-algebras have p-adic operator algebraic analogues, including group(oid) algebras of suitable topological groupoids, noncommutative tori, crossed products, etc. Next, we discuss the construction of a universal p-adic enveloping algebra from a given involutive Banach algebra. If time permits, I will also discuss certain K-theoretic computations. This is joint work with Alcides Buss and Luiz Felipe Garcia.