Leavitt path algebras of weighted and separated graphs
math.RA
/ Authors
/ Abstract
In this paper we show that Leavitt path algebras of weighted graphs and Leavitt path algebras of separated graphs are intimately related. We prove that any Leavitt path algebra $L(E,ω)$ of a row-finite vertex weighted graph $(E,ω)$ is $*$-isomorphic to the lower Leavitt path algebra of a certain bipartite separated graph $(E(ω),C(ω))$. For a general locally finite weighted graph $(E, ω)$, we show that a certain quotient $L_1(E,ω)$ of $L(E,ω)$ is $*$-isomorphic to an upper Leavitt path algebra of another bipartite separated graph $(E(w)_1,C(w)^1)$. We furthermore introduce the algebra $L^{\mathrm{ab}} (E,w)$, which is a universal tame $*$-algebra generated by a set of partial isometries. We draw some consequences of our results for the structure of ideals of $L(E,ω)$, and we study in detail two different maximal ideals of the Leavitt algebra $L(m,n)$.