$C^*$-algebras generated by the paths semigroup

TL;DR

This paper investigates the structure of $C^*$-algebras generated by path semigroup representations on posets, constructing extensions where these algebras become ideals with quotients isomorphic to the Cuntz algebra.

Contribution

It introduces a novel analysis of $C^*$-algebras from path semigroups on posets and constructs extensions linking them to Cuntz algebras.

Findings

Identified the structure of the $C^*$-algebra generated by path semigroup representations.

Constructed algebra extensions where the original algebra is an ideal.

Demonstrated that quotient algebras are isomorphic to the Cuntz algebra.

Abstract

In this paper we study the structure of the $C^*$-algebra, generated by the representation of the paths semigroup on a partially ordered set (poset) and get the net of isomorphic $C^*$-algebras over this poset. We construct the extensions of this algebra, such that the algebra is an ideal in that extensions and quotient algebras are isomorphic to the Cuntz algebra.