Injective spaces of real-valued functions with the baire property

Generalizing the technique used by S.A. Argyros in [3], we give a lemma from which certain Banach spaces are shown to be non-injective. This is applied mainly to study the injectivity of spaces of real-valued Borel functions and functions with the Baire property on a topological space. The results obtained in this way do not follow from previous works about this matter.

Linear extension operators on products of compact spaces

Abstract Let X and Y be the Alexandroff compactifications of the locally compact spaces X and Y , respectively. Denote by Σ( X × Y ) the space of all linear extension operators from C(( X × Y )⧹(X×Y)) to C(( X × Y )) . We prove that X and Y are σ -compact spaces if and only if there exists a T∈Σ( X × Y ) with ‖ T ‖ Γ∈Σ( X × Y ) with ‖ Γ ‖=1. Assuming the existence of a T∈Σ( X × Y ) with ‖ T ‖ X and Y is equivalent to the fact that ‖ Γ ‖⩾2 for every Γ∈Σ( X × Y ) .

Some Problems on Homomorphisms and Real Function Algebras

In this paper we solve a problem about the representation of all homomorphisms on a real function algebra as point evaluations and another two about function algebras in which homomorphisms are point evaluations on sequences in the algebra.

A note on lower bounds of norms of averaging operators

For any natural number n we obtain some examples of continuous onto maps $\phi : S\,\,\longrightarrow\, \,T$ for which Ditor's set $\Delta _\phi ^2(2, 2)$ is empty but every averaging operator for $\phi $ has norm greater or equal to 2n + 1.

On linear extension operators from growths of compactifications of products

Abstract We obtain some results on product spaces. Among them we prove that for noncompact spaces X 1 and X 2 , the norm of every linear extension operator from C ( β ( X 1 × X 2 ) β ( X 1 × X 2 )) into C ( β ( X 1 × X 2 )) is greater or equal than 2, and also that β ( X 1 × X 2 ) β ( X 1 × X 2 ) is not a neighborhood retract of β ( X 1 × X 2 ).

OnK 0-functions and regular extension operators

On the structure of positive homomorphisms on algebras of real-valued continuous functions

In this paper we study the structure of positive homomorphisms on real function algebras. We prove that every positive homomorphism is completely characterized by a family of sets and when the algebra is inverse-closed, by an ultrafilter of zero-sets of functions of the algebra. We show that the known sufficient conditions for every homomorphism of a real function algebra to be countably evaluating or a point evaluation are not necessary. Our results enable us to characterize the countably evaluating algebras as well as the Lindelof spaces as the spaces in which for every algebra, each countably evaluating homomorphism is a point evaluation.

On almost Dugundji spaces and dyadic spaces

Retractive product spaces

Abstract A completely regular Hausdorff space X is called retractive if there is a retraction from βX onto βX \ X . A product space is retractive if and only if all factors are compact but one which is retractive.