ON THE ALGEBRA OF IDEALS AND MODULES IN OPERATOR SPACES

Abstract

The Theory of Operator Ideals and Frechet Modules are important in the study of locally convex spaces, Rings and Algebras. Locally convex spaces are examples of topological vector spaces which generalize normed spaces, so they are Frechet in nature. The original idea in this line was meant to get the interplay between the operator spaces and their subspaces which exhibit the ideal properties from the algebraic point of view. The well known ideals have got certain restrictions on the projections in the spaces, their duals and annihilators. The M−embedments, one- sided structures, multipliers and related theories of r, l−ideals were developed with a hope to enrich the non-commutative attributes and a generalization of ideal structures to specified operator spaces and the clarity of Algebras of operators on Banach spaces, as well as homomorphisms thereof. Despite the fact that studies concerning the Algebra of Ideals and Modules in operator spaces with applications is still active, their general classification and extension remain unsettled. Therefore, the main objective of this study was to characterize the Algebra of Ideals and Modules in certain Operator Spaces. To achieve the objective, we determined the classification of ideals in the set of operators in Banach spaces, characterized the spaces of ideal operators and ideal extensions to Frechet spaces, extended the approximation properties of the ideals through the Integral and Nuclear Operators and determined the algebra of Banach Modules and Functors over the Frechet Spaces. The study employed the methods independently proposed by Godfrey,Kalton, and Saphar, and Sonia to characterize the operator ideals. The hypocontinuity criterion of the Module functors in the Frechet spaces followed the methods proposed by Rieffel. The results demonstrate the existence of classes of closed operator ideals depicting the boundedness in view of Radon- Nikodym properties. Additionally, the findings give the characteristics of ideals through the Hahn-Banach extension operators as well as the necessary and sufficient conditions for the existence of u, h−ideals and their variants in generalized Banach spaces. Finally, the relationship between categorical products and co-products of kernels from one module to the other, flatness of the Module Tensors, and the fact that given an interactive system of modules in a bounded Banach algebra, all canonical morphisms from the module to the collections of its isometric immersions is an isomorphism have been determined. Further characteristics of Frechet modules including; strong factorization properties over the functors, continuity and hypo-continuity of the multiplication have been determined. The findings of this research are significant because of the topological interplay between ideals and modules which allows the hereditary properties of ideals to be used to study modules and opens an area of interest in Algebra. Furthermore, the results display the interplay between Algebra and Analysis, hence contributing to the body of knowledge in both disciplines.