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# Journal of Operator Theory

Volume 60, Issue 2, Fall 2008  pp. 253-271.

Corners of graph algebras

Authors:  Tyrone Crisp
Author institution:

Summary: It is known that given a directed graph $E$ and a subset $X$ of vertices, the sum $\sum\limits_{v\in X}P_v$ of vertex projections in the $C^*$-algebra of $E$ converges (strictly, in the multiplier algebra) to a projection $P_X$. Here we give a construction which, in certain cases, produces a directed graph $F$ such that $C^*(F)$ is isomorphic to the corner $P_X C^*(E)P_X$. Corners of this type arise naturally as the fixed-point algebras of discrete coactions on graph algebras related to labellings. We prove this fact, and show that our construction is applicable to such a case whenever the labelling satisfies an analogue of Kirchhoff's voltage law.

Keywords:  Graph algebra, labelling of a directed graph, fixed-point algebra of a discrete coaction.