Hilbert \(C^{*}\)-modules in which all relatively strictly closed submodules are complemented

We give the characterization and description of all full Hilbert modules and associated algebras having the property that each relatively strictly closed submodule is orthogonally complemented. A strict topology is determined by an essential closed two-sided ideal in the associated algebra and a related ideal submodule. It is shown that these are some modules over hereditary algebras containing the essential ideal isomorphic to the algebra of (not necessarily all) compact operators on a Hilbert space. The characterization and description of that broader class of Hilbert modules and their associated algebras is given. As auxiliary results we give properties of strict and relatively strict submodule closures, the characterization of orthogonal closedness and orthogonal complementing property for single submodules, relation of relative strict topology and projections, properties of outer direct sums with respect to the ideals in \(\ell_\infty\) and isomorphisms of Hilbert modules, and we prove some properties of hereditary algebras and associated hereditary modules with respect to the multiplier algebras, multiplier Hilbert modules, corona algebras and corona modules.

Characterizations of continuous operators on $$C_b(X)$$ with the strict topology

Let X be a completely regular Hausdorff space and $$C_b(X)$$ C b ( X ) be the space of all bounded continuous functions on X, equipped with the strict topology $$\beta $$ β . We study some important classes of $$(\beta ,\Vert \cdot \Vert _E)$$ ( β , ‖ · ‖ E ) -continuous linear operators from $$C_b(X)$$ C b ( X ) to a Banach space $$(E,\Vert \cdot \Vert _E)$$ ( E , ‖ · ‖ E ) : $$\beta $$ β -absolutely summing operators, compact operators and $$\beta $$ β -nuclear operators. We characterize compact operators and $$\beta $$ β -nuclear operators in terms of their representing measures. It is shown that dominated operators and $$\beta $$ β -absolutely summing operators $$T:C_b(X)\rightarrow E$$ T : C b ( X ) → E coincide and if, in particular, E has the Radon–Nikodym property, then $$\beta $$ β -absolutely summing operators and $$\beta $$ β -nuclear operators coincide. We generalize the classical theorems of Pietsch, Tong and Uhl concerning the relationships between absolutely summing, dominated, nuclear and compact operators on the Banach space C(X), where X is a compact Hausdorff space.

Nuclear operators on Cb(X, E) and the strict topology

LetXbe a completely regular Hausdorff space,EandFbe Banach spaces. LetCb(X,E) be the space of allE-valued bounded, continuous functions onX, equipped with the natural strict topologyβ. We study nuclear operatorsT:Cb(X,E) →Fin terms of their representing operator-valued Borel measures.