Involutive operator algebras

TL;DR

This paper develops the general theory of involutive operator algebras, exploring their properties and applications across various mathematical contexts including noncommutative geometry and classical function algebras.

Contribution

It provides a comprehensive framework for involutive operator algebras and demonstrates their relevance in multiple areas of mathematics.

Findings

Characterization of involutive operator algebras

Applications to noncommutative differential geometry

Connections with classical and matrix operator algebras

Abstract

Examples of operator algebras with involution include the operator $*$-algebras occurring in noncommutative differential geometry studied recently by Mesland, Kaad, Lesch, and others, several classical function algebras, triangular matrix algebras, (complexifications) of real operator algebras, and an operator algebraic version of the {\em complex symmetric} operators studied by Garcia, Putinar, Wogen and others. We investigate the general theory of involutive operator algebras, and give many applications.