Almost uniform convergence on the predual of a von Neumann algebra and an ergodic theorem

Abstract

Publisher Summary This chapter discusses almost uniform convergence on the predual of von Neumann algebra and an ergodic theorem. Because L ∞ spaces over measure spaces are the prototypes of commutative von Neumann algebras, sometimes the investigations around a pair ( M , ω ) consisting of a von Neumann algebra M and a positive linear functional on M are called “noncommutative measure” (integration or probability) theory. In addition to the usual topological convergences, these theories need other notions such as almost sure convergence and convergence in measure. The key to the noncommutative generalization is the Egoroff theorem. The chapter focuses on the predual of von Neumann algebra corresponding to a noncommutative L 1 space. M denotes a von Neumann algebra with a faithful normal state ω . The predual M * of M consists of all bounded ultraweakly continuous functionals on M . The proof of individual eryodic theorem in a general form is presented in the chapter.