Algebraization of Operator Theory

Abstract

I have been working in algebra and ring theory, in particular with rings of operators, involutive rings, Baer *-rings and Leavitt path algebras. These rings were introduced in order to simplify the study of sometimes rather cumbersome operator theory concepts. For example, a Baer *-ring is an algebraic analogue of an AW*-algebra and a Leavitt path algebra is an algebraic analogue of a graph C*-algebra. Such rings of operators can be studied without involving methods of operator theory. Thus algebraization of operator theory is a common thread between most of the topics of my interest. After some overview of the main ideas of such algebraization, I will focus on one common aspect of some of the rings of operators – the existence of a trace as a way to measure the size of subspaces/subalgebras. In particular, we adapt some desirable properties of a complex-valued trace on a C*-algebra to a larger class of algebras.