Advanced Studies in Pure Mathematics
- Adv. Stud. Pure Math.
- Operator Algebras and Applications, H. Kosaki, ed. (Tokyo: Mathematical Society of Japan, 2004), 97 - 106
Stable rank and real rank of graph $C^*$-algebras
For any row finite directed graph $E$ there exists a universal $C^*$-algebra $C^*(E)$ ([KPR, KPRR]) generated by projections and partial isometries satisfying the Cuntz-Krieger $E$-relations. This class of graph algebras includes the Cuntz-Krieger algebras and all AF algebras up to stable isomorphisms([D]). In this paper we give conditions for $E$ under which the algebra $C^*(E)$ has stable rank one or real rank zero. A simple graph $C^*$-algebra is either AF or purely infinite, hence it is always extremally rich. We discuss the extremal richness of some graph $C^*$-algebras and present several examples of prime ones with finitely many closed ideals.
First available in Project Euclid: 1 January 2019
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Primary: 46L05: General theory of $C^*$-algebras
Jeong, Ja A. Stable rank and real rank of graph $C^*$-algebras. Operator Algebras and Applications, 97--106, Mathematical Society of Japan, Tokyo, Japan, 2004. doi:10.2969/aspm/03810097. https://projecteuclid.org/euclid.aspm/1546368827