Principal Ideals in Subalgebras of Groupoid C*-Algebras

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.