Presentations of $AF$ Algebras Associated to $T$-Graphs

Abstract

An AF algebra is an inductive limit of finite dimensional C* -algebras Fn, and an embedding of Fn in Fn+l is represented by a graph whose edges are the (multiplicity of the) embeddings of the simple factors of Fn in those of Fn+l (see [1,3]). Thus from a graph F, with distinguished vertex *5 we can build up a unital AF algebra A(F), by iteration of embeddings represented by F, but starting with the complex number C at *. The space of semi-infinite paths F in F beginning at * will be the graph of a Bratteli diagram for A(F). Suppose Fis locally finite, and let F and F denote the vertices and edges respectively of F, and A the incidence matrix of F. We write ||F|| = || ||. A Markov trace Tr on A(F) is given by a solution (0v: v e F ) > 0, y> 0 to