Journal of Operator Theory

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

Corners of graph algebras

Authors: Tyrone Crisp

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.

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