K*-algebras and duality

TL;DR

This thesis extends classical duality theories like Gelfand and Pontryagin duality to broader categories of locally compact spaces and groups, including non-commutative cases, through the study of topological involutive algebras.

Contribution

It generalizes duality theorems to all separated locally compact spaces and groups, including non-commutative structures, using topological involutive algebras.

Findings

Generalized Gelfand duality to all separated locally compact spaces.

Extended Pontryagin duality to all separated locally compact groups.

Developed duality theorems for locally compact semigroups and groups.

Abstract

The work is my Ph D thesis (dissertation for obtaining candidate of sciences degree in Russia) fulfilled under direction of D. A. Raikov and defended under supervision of N. Ya. Vilenkin and S. V. Ptchelintsev. In the dissertatin I gave generalizations of Gelfand duality for compact spaces to the category of all separated locally compact spaces, and of Pontryagin duality for separated commutative locally compact groups to the category of {\em all} (commutative or not) separated locally compact groups. Obtaining of these results has been divided into several stages: 1) the study of topological involutive algebras; 2) the construction of dualities for spaces more general than compact ones; 3) the study of tensor products of topological involutive algebras; 4) actually obtaining duality theorems for all separated locally compact semigroups and groups.