Dual Weights on Crossed Products by Groupoid Actions

In [Y2], we carried out the program of realizing the crossed product by a groupoid action as the left von Neumann algebra of a left Hilbert algebra naturally attached to the given "covariant system". As a consequence, it was shown, as in the case of group actions, that, for each faithful normal positive functional q) on the algebra on which the groupoid is acting, there always exists a faithful normal semifinite weight cp on the crossed product, called the dual weight of (JP. (To avoid difficulty, we dealt with a positive functional only, not with a weight). Several expected results were established such as the fact that the modular automorphism group of q> extends that of cp. It is naturally expected at this stage that this dual weight construction could be done also by exhibiting an operator valued weight of the crossed product to the original algebra, as Haagerup showed in [H4]. The purpose of this paper is to show that this philosophy is indeed the case. The main strategy to achieye our goal can be found in [H4]. However, since we know little about Fourier analysis (or harmonic analysis) on a measured groupoid, we need to provide ourselves with relevant information in this direction. For example, to the best of author's knowledge, no one has ever intensively studied the "Plancherel weight" of a measured groupoid. Thus, naturally, little is known about what functions must be qualified to be called positive definite. Moreover, it should be remarked that the unitary operator A(y) itself, where A(-) is the regular representation of a measured groupoid % has no meaning in the crossed product M x^ by an action a of (it is not even a member of M x^), while, in the group case, it is a typical kind of operators that generate the crossed product. This fact makes our argument more difficult than the one in the case of group actions. For example, because of this situation, we need to adopt an approach in §4 which is different from, but still as interesting as, the ones taken in [H4] or [E&S].