Inequalities In Pre-Hilbert C~*-Modules And Its Applications

Functional analysis is the important branch of the modern mathematics. Operator algebra is the core content of Functional analysis, it contains C*-algebra, Von Neumann al-gebra, etc. Kap. generalized the domain of inner product into C*-algebra, then introduced the concept of C*-modules.In this paper, under the foundation of inner product, we study some inequalities and their applications. The main contents are as follows:The first part is introduction. We mainly introduce C*-modules, besides, we also introduce some theorems and knowledge that are referred frequently in this paper.The second part, we give triangle inequalities, Dunkl-Williams inequalities and some applications.The third part, we give n elements Bohr inequality in C*-algebras. As applications, we give a new generalization of the Dunkl-Williams type inequalities for absolute value in C*-algebras and Hilbert C*-modules and get some inequalities on △A,B(X).The forth part, we give a Cauchy-Schwarz inequality, and discuss the related operator inequalities on semi-inner product modules associated to left multiplier φ, for example Covariance-variance inequality, Wielandt inequality and Kantorovich inequality.