Infinite sum relations on universal C*-algebras

TL;DR

This paper generalizes the theory of universal C*-algebras to include relations involving infinite sums, enabling new descriptions of complex algebras like the Cuntz algebra and Exel-Laca algebras.

Contribution

It introduces a framework for universal C*-algebras with relations defined via the strong operator topology, including infinite sum relations, and provides conditions for norm-based descriptions.

Findings

Described Cuntz algebra of infinite isometries using infinite sum relations

Extended universal property to include strong operator topology relations

Provided conditions for norm relations in projection and partial isometry generated algebras

Abstract

We extend the usual theory of universal C*-algebras from generators and relations in order to allow some relations to be described using the strong operator topology. In particular, we can allow some infinite sum relations. We prove a universal property for the algebras we define and we show how the Cuntz algebra of infinite isometries as well as the Exel-Laca algebras can be described using infinite sum relations. Finally, we give some sufficient conditions for when a C*-algebra generated by projections and partial isometries is a universal C*-algebra using only norm relations, in case one still wants to avoid using relations with respect to the strong operator topology.