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

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 $\cs$ $C^{\ast }(G)$. A corresponding result for a locally compact groupoid $G$ has been investigated by Renault, Ramsay and Walter. We show that the Fourier-Stieltjes algebra $B_{\mu }(G)$ of $G$ (with respect to a quasi-invariant measure $\mu$ on the unit space $X$ of $G$) can be characterized in operator space terms as the dual of the Haagerup tensor product $\ov{L^{2}(X,μ)}^{r}\otimes_{hA}C^{*}(G,μ)\otimes_{h A}L^2(X,μ)^c$ and as the space of completely bounded bimodule maps $C{B}_A(C^{\ast }(G,\mu ),B(L^2(X,\mu )))$, where $A=C_0(X)$ and $C^{\ast }(G,\mu )$ is the groupoid $\cs$ obtained from those $G$-representations associated with $\mu$. 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.

arXiv comment29 pages

References in corpus (1)

• Renault's Equivalence Theorem for Groupoid Crossed Products