A Cuntz--Krieger uniqueness theorem for semigraph $C^*$-algebras

TL;DR

This paper introduces higher rank semigraph algebras, combining ultragraph and higher rank graph algebra concepts, and proves Cuntz--Krieger uniqueness theorems for these structures.

Contribution

It develops a new class of semigraph algebras and establishes foundational uniqueness theorems extending ultragraph algebra results.

Findings

Proved Cuntz--Krieger uniqueness theorems for cancelling semigraph algebras.

Established aperiodic full semigraph algebras.

Extended ultragraph algebra theory to higher rank structures.

Abstract

Higher rank semigraph algebras are introduced by mixing concepts of ultragraph algebras and higher rank graph algebras. This yields a kind of higher rank generalisation of ultragraph algebras. We prove Cuntz--Krieger uniqueness theorems for cancelling semigraph algebras and aperiodic full semigraph algebras.