The ideal structure of C*-algebras of etale groupoids with isotropy groups of local polynomial growth

2024/12/16 by Christensen, Johannes, Neshveyev, Sergey
#FOS: Mathematics #Operator Algebras (math.OA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2412.11805

Abstract

Given an amenable second countable Hausdorff locally compact étale groupoid \mathcal G such that each isotropy group \mathcal G^x_x has local polynomial growth, we give a description of Prim C^*(\mathcal G) as a topological space in terms of the topology on \mathcal G and representation theory of the isotropy groups and their subgroups. The description simplifies when either the isotropy groups are FC-hypercentral or \mathcal G is the transformation groupoid Γ\ltimes X defined by an action Γ\curvearrowright X with locally finite stabilizers. To illustrate the class of C^*-algebras for which our results can provide a complete description of the ideal structure, we compute the primitive spectrum of SL₃(\mathbb Z)\ltimes C₀(SL₃(\mathbb R)/U₃(\mathbb R)), where U₃(\mathbb R) is the group of unipotent upper triangular matrices.

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