On freely generated semigraph $C^*$-algebras

TL;DR

This paper studies a class of universal $C^*$-algebras linked to $k$-semigraphs, establishing their representations, a uniqueness theorem, and computing their $K$-theory, thus broadening the understanding of Cuntz--Krieger type algebras.

Contribution

It introduces a new class of universal $C^*$-algebras for $k$-semigraphs, providing their representations, a uniqueness theorem, and $K$-theory calculations.

Findings

Presented universal representations of the algebras

Proved a Cuntz--Krieger uniqueness theorem

Computed the $K$-theory of these algebras

Abstract

For special universal $C^*$-algebras associated to $k$-semigraphs we present the universal representations of these algebras, prove a Cuntz--Krieger uniqueness theorem, and compute the $K$-theory. These $C^*$-algebras seem to be the most universal Cuntz--Krieger like algebras naturally associated to $k$-semigraphs. For instance, the Toeplitz Cuntz algebra is a proper quotient of such an algebra.