## Illinois Journal of Mathematics

- Illinois J. Math.
- Volume 46, Number 2 (2002), 357-381.

### 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.

#### Article information

**Source**

Illinois J. Math., Volume 46, Number 2 (2002), 357-381.

**Dates**

First available in Project Euclid: 13 November 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1258136198

**Digital Object Identifier**

doi:10.1215/ijm/1258136198

**Mathematical Reviews number (MathSciNet)**

MR1936924

**Zentralblatt MATH identifier**

1038.47045

**Subjects**

Primary: 46L05: General theory of $C^*$-algebras

Secondary: 46L10: General theory of von Neumann algebras 47L40: Limit algebras, subalgebras of $C^*$-algebras

#### Citation

Krishnan, Srilal. Principal ideals in subalgebras of groupoid $C^*$-algebras. Illinois J. Math. 46 (2002), no. 2, 357--381. doi:10.1215/ijm/1258136198. https://projecteuclid.org/euclid.ijm/1258136198