Support of Virasoro unitarizing measures.
It is studied a unitarizing measure of the Virasoro algebra as a probability measure for which the associated space \(L^2\) contains a closed subspace of holomorphic functionals on which this algebra acts unitarily. It is taken into account that the unitarizing property is equivalent to an a priori given formula of integration by parts, which has been previously computed explicitly [\textit{H. Airault} and \textit{P. Malliavin}, J. Math. Pures Appl., IX. Sér. 80, 627--667 (2001; Zbl 1032.58021)]. In addition, it is shown that the unitarizing measures must be supported by the quotient of the homeomorphism group on the circle by the subgroup \(\text{SL}(2,\mathbb{R})\) of Möbius transformations. The work is stimulated by previous investigations related to the canonical Brownian motion on the group diffeomorphisms of the circle [\textit{P. Malliavin}, C. R. Acad. Sci., Paris, Sér. I, Math. 329, 325--329 (1999; Zbl 1006.60073); \textit{H. Airault} and \textit{J. Ren}, J. Funct. Anal. 196, 395--426 (2000; Zbl 1023.60051)] and to the probability measures for representations of Virasoro algebras [\textit{H. Airault}, C. R., Math., Acad. Sci. Paris 334, 787--792 (2002; Zbl 1008.58025)].
scientific article; zbMATH DE number 1850791