Abstract:
For an ample Hausdorff groupoid $\mathcal{G}$, and the Steinberg algebra $A_R(\mathcal{G})$ with coefficients in the commutative ring $R$ with unit, we describe the centralizer of the subalgebra $A_R(U)$ with $U$ an open closed invariant subset of the unit space of $\mathcal{G}$. In particular, it is shown that the algebra of the interior of the isotropy is indeed the centralizer of the diagonal subalgebra of the Steinberg algebra. This will unify several results in the literature, and the corresponding results for Leavitt path algebras follow.
Keywords:
ample groupoid, Steinberg algebra, centralizer, Leavitt path algebra, diagonal of the Leavitt path algebra, commutative core of the Leavitt path algebra.
Citation:
R. Hazrat, Huanhuan Li, “A note on the centralizer of a subalgebra of the Steinberg algebra”, Algebra i Analiz, 33:1 (2021), 246–253; St. Petersburg Math. J., 33:1 (2022), 179–184
\Bibitem{HazLi21}
\by R.~Hazrat, Huanhuan~Li
\paper A note on the centralizer of a subalgebra of the Steinberg algebra
\jour Algebra i Analiz
\yr 2021
\vol 33
\issue 1
\pages 246--253
\mathnet{http://mi.mathnet.ru/aa1744}
\transl
\jour St. Petersburg Math. J.
\yr 2022
\vol 33
\issue 1
\pages 179--184
\crossref{https://doi.org/10.1090/spmj/1695}
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