P-adic C* algebras

Question

I understand that there is a definition of p-adic Banach algebras and that a significant amount of functional analysis can be developed in the non-archimedean setting. Is there a p-adic version of C*-algebras? If so, is there an analogue of the GNS construction?

Answer 1

There exists a complete theory of non-Archimedean commutative Banach algebras. In particular, there are conditions under which an algebra is isomorphic to the algebra of continuous functions. For the commutative case, they can be seen as the counterparts of the $C^*$ condition. Note that there is no natural involution in the p-adic case. See

V. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, AMS, 1990 (the above conditions are in Corollary 9.2.7);

A. Escassut, Ultrametric Banach algebras, World Scientific, 2003.

For the noncommutative case, very little is known.

Answer 2

Me, Alcides Buss, and Devarshi Mukherjee proposed an answer to this question in this paper.

$\textbf{The idea:}$ Note that via GNS, we can view a complex/real $C^*$-algebra essentially as a closed $*$-subalgebra of the space of bounded operators $B(H)$ for some Hilbert space $H$. The idea is to define a $p$-adic $C^*$-algebra as a normed $*$-algebra over $\mathbb{Z}_p$ that is isometrically isomorphic to a closed $*$-subalgebra of the algebra of bounded operators $B(H)$ for some $p$-adic Hilbert space $H$.

$\textbf{But what is a }p\textbf{-adic Hilbert space?}$ A definition of a $p$-adic Hilbert space was proposed by Prof. Andreas Thom and Anton Claussnitzer in this paper. They define it as follows: For every set $X$, define $\mathbb{Q}_p(X)$ as the set of functions $\xi\colon X \to \mathbb{Q}_p$ such that $|\xi(x)|_p > 1$ for at most finitely many $x \in X$. They define a topology on $\mathbb{Q}_p(X)$ that turns it into a locally compact $\mathbb{Z}_p$-module. They show that $\mathbb{Q}_p(X)$ satisfies various properties similar to real/complex Hilbert spaces, such as having an inner product that induces an analogue of the Riesz representation theorem.

$\textbf{What is }B(\mathbb{Q}_p(X))\textbf{ like?}$ Having a definition of Hilbert space, we can define $B(\mathbb{Q}_p(X))$ as the continuous $\mathbb{Z}_p$-linear maps $T\colon \mathbb{Q}_p(X)\to \mathbb{Q}_p(X)$. These maps always have an adjoint, and we equip $B(\mathbb{Q}_p(X))$ with the operator norm. The Banach $\mathbb{Z}_p$-algebra $B(\mathbb{Q}_p(X))$ is isometrically isomorphic to the space of square matrices indexed by $X$ with $p$-adic coefficients, and in each line and each column, the coefficients go to $0$, equipped with the maximum norm. We then introduce $p$-adic operator algebras as normed $*$-algebras over $\mathbb{Z}_p$ that are isomorphic to closed subalgebras of the unity ball of $B(\mathbb{Q}_p(X))$. We choose the unity ball because these algebras are much better behaved.

$\textbf{Examples and properties of }p\textbf{-adic operator algebras:}$ This is where it gets fun. Completions of group algebras, (étale) groupoid algebras, matrix algebras, algebras of continuous functions are all examples of $p$-adic operator algebras. The category of these objects has all limits and colimits that enable us to construct various new examples from our previous ones. Given any $*$-algebra over $\mathbb{Z}_p$, there is the $p$-adic operator algebra that best approximates it in a precise categorical sense. There are (completed) tensor products and crossed products. We can also do some topological $K$-theory with those algebras and compute explicitly some invariants.

We still did not manage to prove a 'GNS,' and this is certainly our dream theorem.