Exel's Effros–Hahn conjecture for Steinberg algebras

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Let KK be a field, let G\mathcal{G} be an ample groupoid, and let u∈G(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

Ind⁡u(M)=KLu⊗KGuM.\operatorname{Ind}_u(M)=KL_u\otimes_{K\mathcal{G}_u}M.

Here Lu=s−1(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=Ann⁡AK(G)(Ind⁡u(M))I=\operatorname{Ann}_{A_K(\mathcal{G})}(\operatorname{Ind}_u(M))

for some u∈G(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.

References

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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