Twisted Topological Graph Algebras

TL;DR

This paper introduces twisted topological graph algebras, extending existing theories by incorporating 1-cocycles, and establishes fundamental properties including simplicity criteria and ideal structure analysis.

Contribution

It defines twisted topological graph algebras, generalizes prior results, and proves new theorems on their structure and simplicity conditions.

Findings

Proved a stronger version of Vasselli's result.

Described gauge-invariant ideal structure.

Established simplicity equivalence with untwisted algebras.

Abstract

We define the notion of a twisted topological graph algebra associated to a topological graph and a $1$-cocycle on its edge set. We prove a stronger version of a Vasselli's result. We expand Katsura's results to study twisted topological graph algebras. We prove a version of the Cuntz-Krieger uniqueness theorem, describe the gauge-invariant ideal structure. We find that a twisted topological graph algebra is simple if and only if the corresponding untwisted one is simple.