Operator Algebra And Operator Theory On RH Module

In the first chapter of this paper some preliminaries are given. The content is mainly about some basic notions of random functional analysis Guo Tiexin introduced: RN space; a.s. boundedness of random operator and random functional on RN space; RN module and its completion (we call it RB module); RN module and its completion (we call it RB module); RIP space, RIP module and its completion (we separately call them RH space, RH module); moreover, we give the proved Riesz representation theorem about a.s. bounded random linear functional on RH module.In the second chapter, we define random Banach algebra and spectrum of its element, and study its basic properties, and generalize many results in the classical Banach algebra theory to the case in random Banach algebra. In order to give a suitable definition of spectrum, we did much try. If we turn a deaf ear to the property of random variable and use the classical definition of spectrum, thenmany results become badly and even have no regularity. For instance, let us consider L(Q,C), a particular random Banach algebra, inspect a random variablewhere A e a and P(A) = \. This is a self-adjoint element, but if we use the classical definition of spectrum, then its spectrum will not be real. For example, a complex valued random variablewhere A e a and P(A) = j-, it is in it. We initially see the important role the positive measure set plays from this example, which leads us to use the present definition.In the third chapter, we define some notions of random C*-algebra and its normal element, self-adjoint element, projection, unitary element, a.s. bounded random positive linear functional, a.s. bounded below of an operator etc, and prove the spectrum theorem of unitary element and self-adjoint element; as a particular randomC*-algebra B(S), we use theorem 3.2 to transform the problem of spectrum into the problem of a.s. bounded below, and prove the spectrum theorem of self-adjoint operator and positive operator. Moreover, we give the correspondent GNS construction under the case that an a.s. bounded random positive linear functional on a random C*-algebra exists.Some properties of positive operators and orthogonal projection operators on random inner product module have been obtained in the paper [14]. In order to keep on studying the properties of positive operators, we give some inequalities about positive operators on RH module in the fourth chapter. These results will be advantageous to further research in the properties of positive operators on RH module.