Operator Algebras and Mauldin Williams Graphs

Abstract

We describe a method for associating a $C^{\ast }$-correspondence to a Mauldin-Williams graph and show that the Cuntz-Pimsner algebra of this $C^{\ast }$-correspondence is isomorphic to the $C^{\ast }$-algebra of the underlying graph. In addition, we analyze certain ideals of these $C^{\ast }$-algebras. We also investigate Mauldin-Williams graphs and fractal $C^{\ast }$-algebras in the context of the Rieffel metric. This generalizes the work of Pinzari, Watatani and Yonetani. Our main result here is a {}``no go'' theorem showing that such algebras must come from the commutative setting.

14 pages, Latex; Rewrote parts of the introduction and the proof of the first main theorem