| dc.description.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. | en_US |
