Preserving Problems On Positive Definite Cones Of Operator Algebras

The preservation problem on operator algebra has always been an important research object of operator theory and operator algebra.In recent years,many experts and scholars have carried out a series of research on the retention problem on different algebras and achieved fruitful results.For example,Molnar etc.studied the preservation of geometric mean,harmonic mean,arithmetic mean,and power mean on C*-algebra.Influenced and inspired by this,this paper mainly discusses the structure of Heron mean preserving mapping on the positive cone of C*-algebra,and studies the Heron mean preserving mapping on the positive cone of von Neumann algebra.At the same time,Geher characterizes the structure of mappings that preserve absolute continuity and singularity in both directions on operator algebra.In this paper,the structure of preserving absolutely continuous and singularity mappings in both directions on the positive cone of von Neumann algebras is obtained.Chapter 1 mainly introduces the research background and basic knowledge of preserving problems on operator algebras;In Chapter 2,we mainly characterize the mapping preserving Heron mean on the positive definite cone of operator algebra.The first section of Chapter 2 introduces the definition of mean and the basic concepts and main conclusions of Kubo-Ando mean value theory;In Section 2,the necessary and sufficient conditions for preserving Heron mean mapping on the positive cone of C*-algebra are given;In Section 3,it is proved that the mapping satisfying the preservation of Heron mean on von Neumann algebras without direct sum terms of type I1 and type I2 must be constant;In Chapter 3,we mainly characterize the structure of preserve absolutely continuous and singular mappings on von Neumann algebras.In the first section of Chapter 3,we introduce the basic concepts of absolute continuity and singularity and Geher’s main conclusions on maintaining absolute continuity and singularity mappings on positive cones of C*-algebras;In Section 2,We proved that if the bijection Φ preserves absolutely continuous in both directions then it preserves singular in both directions on von Neumann algebras.Moreover,this bijection Φ can be characterized by bounded,invertible,linear or conjugate linear operators of the Hilbert space.