Commun. Korean Math. Soc. 2000 Vol. 15, No. 3, 521-531
J. A. Jeong Seoul National University
Abstract : Graph $C^{*}$-algebras $C^{*}(E)$ are the universal $C^*$-algebras generated by partial isometries satisfying the Cuntz-Krieger relations determined by directed graphs $E$, and it is known that a simple graph $C^*$-algebra is extremally rich in a sense that it contains enough extreme partial isometries in its closed unit ball. In this short paper, we consider a sufficient condition on a graph for which the associated graph algebra (possibly nonsimple) is extremally rich. We also present examples of nonextremally rich prime graph $C^*$-algebras with finitely many ideals and with real rank zero.
