Dual weights on crossed products by groupoid actions (Q1328890)

In Publ. Res. Inst. Math. Sci. Kyoto Univ. 28, 535-576 (1992; review above), the author proved that the crossed product of a groupoid action can be realized as the left von Neumann algebra of a left Hilbert algebra naturally attached to the given ``covariant system''. As a consequence it was shown that for each faithful normal positive functional \(\varphi\) on the algebra on which the groupoid is acting there always exists a faithful normal semifinite weight \(\widetilde \varphi\) on the crossed product called the dual weight of \(\varphi\). In this paper it is shown that this dual weight construction can be done also by exhibiting an operator valued weight of the crossed product to the original algebra. The paper uses the approach of \textit{U. Haagerup}, Math. Scan. 43, 119-140 (1978; Zbl 0405.46053).