Infinite-dimensional diagonalization and semisimplicity

TL;DR

This paper extends the classical theory of diagonalizable operators to infinite-dimensional vector spaces, characterizing diagonalizable subalgebras using a natural topology and generalizing semisimplicity concepts.

Contribution

It provides a new characterization of diagonalizable subalgebras in infinite dimensions and generalizes the Wedderburn-Artin theorem to infinite-dimensional rings.

Findings

Characterization of diagonalizable subalgebras in End(V) using the finite topology.

Conditions for simultaneous diagonalization of subalgebras.

Infinite-dimensional generalization of the Wedderburn-Artin theorem.

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.