Diagonalizable algebras of operators on infinite-dimensional vector spaces

Given a vector space V over a field K, let End(V) denote the algebra of linear endomorphisms of V. If V is finite-dimensional, then it is well-known that the diagonalizable subalgebras of End(V) are characterized by their internal algebraic structure: they are the subalgebras isomorphic to K^n for some natural number n. 

In case V is infinite dimensional, the diagonalizable subalgebras of End(V) cannot be characterized purely by their internal algebraic structure: one can find diagonalizable and non-diagonalizable subalgebras that are isomorphic.  I will explain how to characterize the diagonalizable subalgebras of End(V) as topological algebras, using a natural topology inherited from End(V).  I will also illustrate how this characterization relates to an infinite-dimensional Wedderburn-Artin theorem that characterizes "topologically semisimple" algebras.