Identification of maximal $C^*$-covers of some operator algebras

TL;DR

This paper determines the maximal C*-envelope of certain operator algebras, specifically upper triangular matrices and related graph algebras, using free product and completely positive map techniques.

Contribution

It extends the understanding of maximal C*-covers for upper triangular matrix algebras and graph algebras through new analytical methods.

Findings

Identified maximal C*-envelopes for 3x3 upper triangular matrices

Extended results to larger upper triangular matrices

Applied techniques to graph algebras of cycle graphs

Abstract

We use results on inclusions of free products and extensions of completely positive maps to determine the maximal $C^*$-envelope for upper triangular $3 \times 3$ matrices. We consider these same results in the context of larger upper triangular matrices and graph algebras associated to cycle graphs.