Applications of von Neumann Algebras to Rigidity Problems of (2-step) Riemannian (Nil-)Manifolds

Hamid-Reza Fanaï；Atefeh Hasan-Zadeh

Von Neumann algebras ； 2-step nilmanifolds ； Free and ergodic actions ； Derivations ； Automorphisms

Tamsui Oxford Journal of Information and Mathematical Sciences

33卷1期（2019 / 07 / 01）

56 - 61

英文

In this paper, basic notions of von Neumann algebra and its direct analogous in the realm of groupoids and measure spaces have been considered. By recovering the action of a locally compact Lie group from a crossed product of a von Neumann algebra, other proof of one of a geometric proposition of O’Neil and an extension of it has been proposed. Also, using the advanced exploration of nilmanifolds in measure spaces and their corresponding automorphisms (Lie algebraic derivations) a different proof of an analytic proposition of Gordon and Mao has been attained. These two propositions are of the most important ones for rigidity problems of Riemannian manifolds especially 2-step nilmanifolds.