Ergodic Theorems For Noncommutative Lorentz Spaces

The research of non-commutative Lp spaces become to mature, many researchers began to study the theory of non-commutative Lorentz spaces. In 1981, H.Kosaki gave the definition and properties of Lorentz space for the case p≥,q≥1. In recent years, the research results of Q.Xu, V.Chilin etc, make the theory of non commutative Lorentz space more and more rich. This thesis will discuss individual ergodic theorems in non-commutative Lorentz space associated with a semifinite von Neumann algebra. The thesis is organized as follows:In the introduction we briefly state the background and the main results of this thesis.In Chapter 1, we introduce some notations, definitions, properties of operators, basic knowledge of non-commutative Lp-spaces and singular values of τ-measurable opera-tors.The second chapter is the main part of this thesis. It is divided into two sections. Sec-tion 1 gives definitions, properties of the non-commutative Lorentz spaces, and introduces almost uniform convergence (bilateral uniform convergence), uniformly equicontinuous in measure (bilateral uniformly equicontinuous in measure) and related results. Section 2 we prove individual ergodic theorems in the noncommutative Lorentz space Lp,q(M).