Fourier–Stieltjes Algebras of Locally Compact Groupoids

Abstract

For locally compact groups, Fourier algebras and Fourier–Stieltjes algebras have proven to be useful dual objects. They encode the representation theory of the group via the positive definite functions on the group: positive definite functions correspond to cyclic representations and span these algebras as linear spaces. They encode information about the algebra of the group in the geometry of the Banach space structure, and the group appears as a topological subspace of the maximal ideal space of the algebra. Because groupoids and their representations appear in studying operator algebras, ergodic theory, geometry, and the representation theory of groups, it would be useful to have a duality theory for them. This paper gives a first step toward extending the theory of Fourier–Stieltjes algebras from groups to groupoids. IfGis a locally compact (second countable) groupoid, we show that B(G), the linear span of the Borel positive definite functions onG, is a Banach algebra when represented as an algebra of completely bounded maps on aC*-algebra associated withG. This necessarily involves identifying equivalent elements of B(G). An example shows that the linear span of the continuous positive definite functions need not be complete. For groups, B(G) is isometric to the Banach space dual ofC*(G). For groupoids, the best analog of that fact is to be found in a representation of B(G) as a Banach space of completely bounded maps from aC*-algebra associated withGto aC*-algebra associated with the equivalence relation induced byG. This paper adds weight to the clues in earlier work of M. E. Walter on Fourier–Stieltjes algebras that there is a much more general kind of duality for Banach algebras waiting to be explored.