The Fourier algebra of a measured groupoid and its multipliers (Q1355077)

The paper extends many basic results about Fourier algebra \(A(G)\), Fourier-Stieltjes algebra \(B(G)\) and their multiplier spaces to the case of measured groupoids \(G\). For this purpose the author defines analogues of positive definite functions (called positive type functions on a groupoid) and proves basic properties of them. Next, the definitions of \(B (G)\) and \(A(G)\) are given, and properties of those spaces are established, in analogy with the group case. These include, among others, proving the fact that \(B(G)\) is a Banach algebra containing \(A(G)\) as a closed involutive ideal. Then the author exhibits the space \(C^*_\mu (G) \) -- the full \(C^*\)-algebra of a measured groupoid \(G\) -- and uses it to construct the predual of \(B(G)\) as a Haagerup tensor product with \(L^2(G^{(0)})\). He also defines the space \(VN(G)\) -- the von Neumann algebra of \(G\) -- and constructs the dual of \(A(G)\) which turns out to be a Haagerup tensor product of \(VN(G)\) and \(L^2 (G^{(0)})\), and proves the respective dualities. Then the author considers the multiplier algebra \(MA(G)\) and the algebra \(M_0 A(G)\) of completely bounded multipliers of the Fourier algebra \(A(G)\) of a measured groupoid. A definition of amenable groupoids is given, with several equivalent conditions of amenability. It is proved that if a groupoid \(G\) is amenable, then \(MA(G) =B(G)\). The author also gives a definition of Herz-Schur multipliers of a groupoid as invariant elements of \(B(G^*G)\), and shows its equivalence with completely bounded multipliers. Finally, the author extends results of Pisier and Varopoulos on Littlewood functions to the case of groupoids. The main result about those is that the space of Littlewood functions on a measured groupoid coincides with the space of multipliers from \(L^\infty (G)\) into \(B(G)\), with equivalence of the respective norms. It is also proved that the absolute Fourier multipliers of \(G\) coincide with Varopoulos functions on \(G\).