Principal ideals in subalgebras of groupoid $C^*$-algebras

Abstract

The study of different types of ideals in non self-adjoint operator algebras has been a topic of recent research. This study focuses on principal ideals in subalgebras of groupoid $C^*$-algebras. An ideal is said to be principal if it is generated by a single element of the algebra. We look at subalgebras of $r$-discrete principal groupoid $C^*$-algebras and prove that these algebras are principal ideal algebras. Regular canonical subalgebras of almost finite $C^*$-algebras have digraph algebras as their building blocks. The spectrum of almost finite $C^*$-algebras has the structure of an $r$-discrete principal groupoid and this helps in the coordinization of these algebras. Regular canonical subalgebras of almost finite $C^*$-algebras have representations in terms of open subsets of the spectrum for the enveloping $C^*$-algebra. We conclude that regular canonical subalgebras are principal ideal algebras.