Three-space Property Research Articles

Twisted Hilbert spaces

A Banach space X is called a twisted sum of the Banach spaces Y and Z if it has a subspace isomorphic to Y such that the corresponding quotient is isomorphic to Z. A twisted Hilbert space is a twisted sum of Hilbert spaces. We prove the following tongue-twister: there exists a twisted sum of two subspaces of a twisted Hilbert space that is not isomorphic to a subspace of a twisted Hilbert space. In other words, being a subspace of a twisted Hilbert space is not a three-space property.

Some properties of locally quasi-convex groups

We prove in this paper that for a Hausdorff group topology on an Abelian group with sufficiently many continuous characters, there is an associated locally quasi-convex topology which is the strongest among all the locally quasi-convex group topologies weaker than the given one. We also give a result on local quasi-convexity on the line of three-space properties.

The Dunford-Pettis property is not a three-space property

An example of a Banach spaceX is shown which does not have the Dunford-Pettis property, while some subspaceY and the corresponding quotientX/Y have the hereditary Dunford-Pettis property.

Nuclear Fr�chet spaces with basis do not have the three-space property

A proper ty (P) that locally convex spaces may enjoy is termed a three-space property if, whenever E contains a subspace ( = closed subspace) F such that F and ElF have (P), then E too has (P). In many situations it is important to know whether a given property (P) is a three-space property. Here we shall prove that the property of having a basis is not a three-space proper ty for nuclear Fr6chet spaces. This will follow from showing that the proper ty of being not twisted is not a three-space property and from the result in [2; w 2].