On gω-Open Sets in Grill Topological Spaces

The propose of this paper is to introduce and investigate a weak form of ω-open set in grill topological spaces. We introduce the notion of -open set as a form stronger than βω-open set and weaker than ω-open set and -open set. By using this form, we study the generalization property, the interior operator, closure operator and θ-cluster operator.

Completions of Boolean algebras of projections and weak-star closures of C*-algebras on dual Banach spaces

Palmer has shown that those hermitians in the weak-star operator closure of a commutative C*-algebra represented on a dual Banach space X that are known to commute with the initial C*-algebra form the real part of a weakly closed C*-algebra on X. Relying on a result of Murphy, it is shown in this paper that this last proviso may be dropped, and that the weak-star closure is even a W*-algebra.When the dual Banach space X is separable, one can prove a similar result for C*-equivalent algebras, via a ‘separable patch’ completion theorem for Boolean algebras of projections on such spaces.

Boolean algebras of projections and resolutions of the identity of scalar-type spectral operators

Let Μ be a Bade complete (or σ-complete) Boolean algebra of projections in a Banach space X. This paper is concerned with the following questions: When is Μ equal to the resolution of the identity (or the strong operator closure of the resolution of the identity) of some scalar-type spectral operator T (with σ(T) ⊆ ℝ) in X? It is shown that if X is separable, then Μ always coincides with such a resolution of the identity. For certain restrictions on Μ some positive results are established in non-separable spaces X. An example is given for which Μ is neither a resolution of the identity nor the strong operator closure of a resolution of the identity.