Exel's Effros–Hahn conjecture for Steinberg algebras

Let KK be a field, let G\mathcal{G} be an ample groupoid, and let uG(0)u\in\mathcal{G}^{(0)}. Write AK(G)A_K(\mathcal{G}) for the Steinberg algebra, Gu\mathcal{G}_u for the isotropy group at uu, and, for a left KGuK\mathcal{G}_u-module MM, write

Indu(M)=KLuKGuM.\operatorname{Ind}_u(M)=KL_u\otimes_{K\mathcal{G}_u}M.

Here Lu=s1(u)L_u=s^{-1}(u) and II is a primitive ideal of AK(G)A_K(\mathcal{G}). Exel's Effros–Hahn conjecture. Then

I=AnnAK(G)(Indu(M))I=\operatorname{Ann}_{A_K(\mathcal{G})}(\operatorname{Ind}_u(M))

for some uG(0)u\in\mathcal{G}^{(0)} and some simple left KGuK\mathcal{G}_u-module MM. The conjecture is the Steinberg-algebra analogue of the Effros–Hahn conjecture, which asserts that primitive ideals arise by induction from isotropy data. The source records this as a conjecture proposed by R. Exel at the 2014 PARS meeting; its status is not established in the supplied text.

Sources & referencesView supporting material

Primary source

T. T. H. Duyen, D. Gonçalves and T. G. Nam, “On the ideals of ultragraph Leavitt path algebras”, arXiv:2109.10440 (2021).

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