Maps On Positive Cones In Operator Algebras Preserving Means

The study of means is one of the hot topics in operator algebras which has aroused the interest of many scientists and mathematicians in recent years.The most important concepts of the means in the positive cones of operator algebras comes from Kubo and Ando.In the following years,the means as an important tool greatly promoted the development of operator algebras.In chapter one,the development history of means and the elementary knowledge related to it in operator algebras are given.In chapter two,we mainly study the maps in operator algebras which preserve power means and Heron means.In the first section,we introduce the basic concepts and results of the theory of the Kubo-Ando means.In the second section,we give the conclusions for the structure of the maps which preserve power means in operator algebras.In the third section,the definition of p-norm and its related properties are given.We study the general form of the maps between positive cones which preserve the p-norm of conventional weighted power means.Finally,in the fourth section,we will generalize the conclusions of Gaal to the Heron means.We also characterize the order isomorphism using Heron means.The form of the maps that preserve the Heron means are also given.