C~*-algebra Valued Norms And Applications

Norm is a very fundamental and important notion in functional analysis. This thesis consists of two chapters which concern the generalization of the norm to C~*-algebra valued norms on linear spaces.In chapter 1,we define the notion of C~*-algebra valued norms on linear spaces,give some examples and construction,and characterize the property of C~*-algebra valued norms;for a normed linear space,we give some conception such as the C~*-algebra valued norms preserving cauchy sequences,F-cauchy sequences, F-convergence sequences,F-complete,F-Banach space,study their relevant relations; we also prove that the space of all the bounded linear operators is a F-Banach space on the certain conditions.In chapter 2,we generalize the notion of C~*-algebra valued norms to define the L(C(K))-valued norms on a linear space,where L(C(K)) is the algebra of all the bounded linear operators on the continuous functions space C(K) on a compact Haussdorff space K.We prove that every bounded L(C(M_β))-valued norm on a unital abelian semi-simple Banach algebraβis multiplicative.