Maps On Positive Cones Of Operator Algebra To Preserve The Properties Of Spectral Geometric Mean

Mean is an important concept in operator algebras.Common mean values include algebraic mean,geometric mean,harmonic mean,power mean,etc.In 1980,Kubo and Ando proposed the concept of Kubo-Ando mean,one of the most important concept in the mean value problem of operator algebra.Later,S.Kim and H.Lee also studied the quantum relative entropy related to spectral geometric mean.Molnar,Trapp,Gaal and other authors have made a series of studies on the related problems of many means.They got many interesting results and greatly promoted the development of the research fields of means in operator algebra.In chapter 1,we mainly introduce the development of the means in operator algebra.Some basic knowledge and some notations are also given in this chapter.In chapter 2,we study those maps between positive definite cones of operator algebra which preserve the spectral geometric mean.In section 1 of chapter 2,we study the form of the maps which preserves the spectral geometric mean in operator algebra.In section 2 of chapter 2,the definitions of weighted log-euclidean mean and(A1-?/2zB?/zA1-?/2z)z mean are introduced.The concepts of Qs(A||B),Q?le(A||B)Qa,Z(A||B)are also given.We prove that if any of the quanlities Q?le(A||B)and Q?,z(A||B)on one C*-algebra can be transformed to the quantity Qs(A||B)on another C*-algebra by a surjective transformation between the positive definite cones,then those algebras are necessarily commutative.In section 3 of chapter 2,we consider the maps which preserve the unitary invariant norm of the spectral geometric mean.In section 4 of chapter 2,the definition of p-norm and its related properties are given.We then study the general form of the maps between positive definite cones which preserve the p-norm of the spectral geometric means and the geometric means.Finally,in section 5 of chapter 2,we study the structure of local automorphisms of spectral geometric means on B(H)++.