Noncommutative Orlicz Spaces And Some Basic Topological And Geometric Properties

The main aim of this thesis is to study some geometric and topological properties about the noncommutative Orlicz spaces L_?????,??,including the closed subspaces,the uniformly monotone,Kadec-Klee property for convergence in measure,the lower bound estimation of the packing constant,some geometric properties of noncommutative Orlicz sequence spaces and so on.This thesis consists of the following six chapters.In Chapter 1,we present a general survey of the backgrounds and modern developments for noncommutative Orlicz spaces.Subsequently we show the important conclusions of this thesis,and related results for measurable operators and some concepts of noncommutative Orlicz spaces,which will be used throughout this thesis.In Chapter 2,one establishes as a modular space,the space L_?????,??possesses the Fatou property,and consequently,it is a Banach space.In addition,the description of the subspace E_?????,??which is closed under the norm topology and dense under the measure topology,is given.Moreover,if the Orlicz function ? satisfies the ?p??₂ then L_?????,??is uniformly monotone,and convergence in the norm topology and measure topology coincide on the unit sphere.Hence,L_?????,??= E_?????,??i? satisfies the ?₂ condition.At last of this Chapter,some properties of Orlicz norm on L_?????,??are given,such as the relationship of Orlicz norm and modular,the equivalence of convergence in the measure and the norm,absolute continuity of Orlicz norm,at last the Orlicz norm of the projection operator is given.In Chapter 3,we present the Kadec-Klee property for convergence in measure of the noncommutative Orlicz spaces L_?????,??.Firstly,one proves that L_?????,??have the Kadec-Klee property for convergence in measure if and only if? if satisfies the ?₂ condition.Hence,L_?????,??have the property of order continuous and local uniform monotonicity properties.At last of this Chapter,the dual space and reflexivity of L_?????,??are given.In Chapter 4,we mainly study the packing constant.Firstly,one calculate the Kottman constant and the packing constant of the Cesàro-Orlicz sequence space,in order to compute the constants,the paper gives two formulas.On the base of these formulas one can easily obtain the packing constant for the Cesàro sequence space and some other sequence spaces.Secondly,a new constant ????X?which seems to be relevant to the packing constant,is given.At last,similar the classical case we calculate the lower bound of the packing constant of the noncommutative Orlicz spaces L_?????,??,and one proves all of the packing constant of the noncommutative space L₁??,??and L_???,??are2/1.In Chapter 5,we present some results of the noncommutative Orlicz sequence spaces L_??B?H??which generalize the Schatten class.First of all,the definition of noncommutative Orlicz sequence spaces L_??BH?)is given,and some basic properties of this space are presented,such as the relationship between norm and trace,trace inequality and so on,which generalize the results of S₁?H?and S_p?H??1<p<??.In the last part of this Chapter,one gives the criterions of the extreme point and the round space about L_??BH?),as corollary,the corresponding results of S₁?H?and S_p?H??1<p<??are given.In Chapter 6,we summarize the main contribution of this thesis and makes the expec-tation for the further study.