Universal C$^{\ast}$-Algebras from Graph Products: Structure and Applications

TL;DR

This paper introduces a new class of C$^{\

Contribution

It develops a framework for graph product C$^{\ast}$-algebras, revealing their structural properties and applications, including universal properties, nuclearity, and ideal structure analysis.

Findings

Established universal properties of the new C$^{\ast}$-algebras

Characterized nuclearity and exactness in terms of vertex algebras

Analyzed the ideal structure of graph product C$^{\ast}$-algebras

Abstract

In this article, we introduce and investigate a class of C$^{\ast}$-algebras generated by reduced graph products of C$^{\ast}$-algebras, augmented with families of projections naturally associated with words in right-angled Coxeter groups. These ambient C$^{\ast}$-algebras possess a rich and tractable combinatorial structure, which enables the deduction of a variety of structural properties. Among other results, we establish universal properties, characterize nuclearity and exactness in terms of the vertex algebras, and analyze the ideal structure. In the second part of the article, we leverage this framework to derive new insights into the structure of graph product C$^{\ast}$-algebras -- many of which are novel even in the case of free products.