Some topics pertaining to algebras of linear operators

Here we consider finite-dimensional vector spaces and linear operators on them. Some of the basic ingredients will be motivated by ∗-algebras of linear transformations on inner product spaces, as in [Arv, Dou]. In particular, let us note that in the theory of C and Von Neumann algebras, the “double commutant” of a ∗-algebra is a kind of basic closure of the algebra. A special case concerns algebras of operators generated by representations of finite groups [Bur, Cur2, Ser2]. The use of characters is included in each of these three books, and is generally avoided in the present article. Part of the point of view about the p-adic absolute value and p-adic numbers that we follow here is that they can be interesting in connection with theory of computation and related settings just because of their nice properties and simplicity, aside from more traditional number-theoretic uses, even if those are also very interesting.