Riemannian Metrics On The C*-algebras Equipped With Actions Of Compact Groups

We study the Riemannian metrics on the C~*-algebras equipped with actions of compact groups and some properties of them in this dissertation.We introduce,in Chap-ter 1,the background,basic definitions and basic results of this dissertation.In Chapter2 we construct calculi with quasi-correspondences on the C~*-algebras equipped with actions of compact groups,and then we obtain the corresponding qCdCs;We also give a sufficient condition for the calculi with quasi-correspondences to be Riemannian met-rics.In Chapter 3 we prove that energy forms on the C~*-algebras equipped with com-pact ergodic actions satisfy the Markov property and the corresponding energy norms have strongly Leibniz property;We obtain a Laplace operator from a Riemannian metric on a C~*-algebra equipped with compact ergodic action,and then we construct a CdC on this C~*-algebra by the Laplace operator;We give a condition for the CdC to be the original CdC.In Chapter 4 we define an energy metric on the state space of the C~*-algebra from an energy form;We prove that there are affine isometries from the state spaces(as metric spaces)into some Hilbert spaces;We also describe the metrics using the Laplace operators on the C~*-algebras.In Chapter 5 we obtain the general forms of CdCs on a class of the C~*-algebras equipped with actions of compact groups.