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Using the Steinberg algebra model to determine the center of any Leavitt path algebra. (English) Zbl 1469.16063

The authors compute the center of any Leavitt path algebra \(L_R(E)\) of a directed graph \(E\) over a commutative unital ring \(R\). This result generalizes various results in the literature dealing with the computation of the center of a Leavitt path algebra, where the graph or/and the ground ring \(R\) are assumed to satisfy additional conditions.
The authors use the fact that the Leavitt path algebra \(L_R(E)\) is isomorphic to the Steinberg algebras \(A_R(\mathcal G_E)\) of certain ample Hausdorff groupoid \(\mathcal G_E\), called the graph groupoid of \(E\). Combining this with a result of Steinberg which describes the center of a Steinberg algebra \(A_R(\mathcal G)\) as the set of all class functions in \(A_R(\mathcal G_E)\), the authors are able to provide an explicit description of the center of \(L_R(E)\) in terms of concrete sets of vertices and paths in the graph.
Moreover, they show that the center of a Leavitt path algebra \(L_R(E)\) is a graded ring, and describe its graded ring structure, showing that it is graded-isomorphic to a direct sum of copies of \(R\), in degree \(0\), and of copies of \(R[x,x^{-1}]\), with its usual grading.

MSC:

16S88 Leavitt path algebras
46L05 General theory of \(C^*\)-algebras
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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References:

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