Infinite-dimensional diagonalization and semisimplicity

Miodrag C Iovanov,Zachary Mesyan,Manuel L Reyes

doi:10.1007/s11856-016-1395-5

Abstract

We characterize the diagonalizable subalgebras of End(V), the full ring of linear operators on a vector space V over a field, in a manner that directly generalizes the classical theory of diagonalizable algebras of operators on a finite-dimensional vector space. Our characterizations are formulated in terms of a natural topology (the "finite topology") on End(V), which reduces to the discrete topology in the case where V is finite-dimensional. We further investigate when two subalgebras of operators can and cannot be simultaneously diagonalized, as well as the closure of the set of diagonalizable operators within End(V). Motivated by the classical link between diagonalizability and semisimplicity, we also give an infinite-dimensional generalization of the Wedderburn-Artin theorem, providing a number of equivalent characterizations of left pseudocompact, Jacoboson semisimple rings that parallel various characterizations of artinian semisimple rings. This theorem unifies a number of related results in the literature, including the structure of linearly compact, Jacobson semsimple rings and cosemisimple coalgebras over a field.

Highlights

Let V be a vector space over a field K, and let End(V ) be the algebra of K-linear operators on V
Through a detailed study of topological algebras of the form KX for some set X, we show that such “function algebras” with continuous algebra homomorphisms form a category that is dual to the category of sets
We study diagonalizable subalgebras that are not necessarily closed and individual diagonalizable operators, still making use of the finite topology on End(V )

Summary

Introduction

Let V be a vector space over a field K, and let End(V ) be the algebra of K-linear operators on V. Under the isomorphism (identification) φ : A → V given by φ(1) = v0 and φ(xi) = vi for i ≥ 1, the proof of Proposition 2.3 provides an injective homomorphism λ : A → End(V ) Such that λ(A) is a closed discrete maximal commutative subalgebra of End(V ). (These remarks apply to limits of general diagrams, but we restrict to inversely directed systems for notational simplicity as these are the only systems we require.) Given a topological ring R, let RTMod denote the category of left topological R-modules with continuous module homomorphisms. Our discussion of the finite topology above makes it clear that the topological ring End(V ) for a right D-vector space V is left pro-discrete. Fixing an infinite-dimensional K-vector space V , to what extent can one characterize those left pro-discrete K-algebras that can be realized as closed subalgebras of End(V )?

Infinite Wedderburn-Artin Theorem

Diagonalizable algebras of operators