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# Journal of Operator Theory

Volume 41, Issue 1, Winter 1999  pp. 175-197.

Fourier-Stieltjes algebras of $r$-discrete groupoids

Authors Karla J. Oty
Author institution: Department of Mathematics, Southeaston Oklahoma State University, Durant, OK 74701, U.S.A.

Summary:  The Fourier-Stieltjes algebra, ${\cal B}(G)$, of a groupoid $G$ has recently been defined using suitably defined positive definite functions. In this paper we prove various properties of positive definite functions including that continuous positive definite functions separate the points of $G$. We also show that in certain cases the continuous elements of ${\cal B}(G)$ (denoted $B(G))$ and the space of complex-valued bounded continuous functions (denoted $C(G))$ are topologically isomorphic as Banach algebras but not as ordered or\break $*$-Banach algebras. The same is shown to be true for ${\cal B}(G)$ and the space of complex-valued bounded Borel functions (denoted $M(G))$. We explore various conditions including that of an ordering map that one can place on groupoids and their connections, and we show that if $G$ has such an ordering map then $C_{\rm c}(G)\subseteq B(G)$ and $M_{\rm c}(G)\subseteq {\cal B}(G)$.

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