An Operator Space duality theorem for the Fourier-Stieltjes algebra of a locally compact groupoid

2011/02/01 by Paterson, Alan L. T.
#43A32 #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1102.0304

Abstract

It is a well-known result of Eymard that the Fourier-Stieltjes algebra of a locally compact group G can be identified with the dual of the group \cs C^*(G). A corresponding result for a locally compact groupoid G has been investigated by Renault, Ramsay and Walter. We show that the Fourier-Stieltjes algebra B_μ(G) of G (with respect to a quasi-invariant measure μ on the unit space X of G) can be characterized in operator space terms as the dual of the Haagerup tensor product \ovL²(X,μ)^r⊗_hAC^*(G,μ)⊗_{h A}L²(X,μ)^c and as the space of completely bounded bimodule maps CB_A(C^*(G,μ),B(L²(X,μ))), where A=C₀(X) and C^*(G,μ) is the groupoid \cs obtained from those G-representations associated with μ. A similar but different result has been given by Renault, but our proof is along different lines, and full details are given. Examples illustrating the result are discussed.

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