Nebbia's CCR conjecture for boundary-transitive groups on semi-regular trees

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Let TT be a thick semi-regular tree, and let GG be a closed subgroup of Aut⁡(T)\operatorname{Aut}(T) acting transitively on the boundary ∂T\partial T. A locally compact group GG is CCR if π(f)\pi(f) is compact for every irreducible representation π\pi of GG and every f∈L1(G)f\in L^1(G). Nebbia's conjecture. Every such group GG is CCR.

The conjecture proposes that sufficiently regular boundary action forces regularity of the unitary dual. Earlier results establish implications from CCR or Type I assumptions to boundary transitivity, while the converse remains open in this generality.

References

Primary source

Lancelot Semal, “Irreducibly represented Lie groups and Nebbia's CCR conjecture on trees”, arXiv:2306.04310 (2023).

Additional references

2 papers in this index state this conjecture (2022–2023). The statement above is taken from the most recent of them; the others are arXiv:2203.04189.

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