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On the classes of hereditarily ℓ p Banach spaces

Parviz Azimi, A. A. Ledari (2006)

Czechoslovak Mathematical Journal

Let X denote a specific space of the class of X α , p Banach sequence spaces which were constructed by Hagler and the first named author as classes of hereditarily ℓ p Banach spaces. We show that for p > 1 the Banach space X contains asymptotically isometric copies of ℓ p . It is known that any member of the class is a dual space. We show that the predual of X contains isometric copies of ℓ q where 1 p + 1 q = 1 . For p = 1 it is known that the predual of the Banach space X contains asymptotically isometric copies of c 0 . Here we...

On the Dunford-Pettis property of tensor product spaces

Ioana Ghenciu (2011)

Colloquium Mathematicae

We give sufficient conditions on Banach spaces E and F so that their projective tensor product E ⊗ π F and the duals of their projective and injective tensor products do not have the Dunford-Pettis property. We prove that if E* does not have the Schur property, F is infinite-dimensional, and every operator T:E* → F** is completely continuous, then ( E ⊗ ϵ F ) * does not have the DPP. We also prove that if E* does not have the Schur property, F is infinite-dimensional, and every operator T: F** → E* is completely...

On the structure of non-dentable subsets of C ( ω ω k )

Pericles D. Pavlakos, Minos Petrakis (2011)

Studia Mathematica

It is shown that there is no closed convex bounded non-dentable subset K of C ( ω ω k ) such that on subsets of K the PCP and the RNP are equivalent properties. Then applying the Schachermayer-Rosenthal theorem, we conclude that every non-dentable K contains a non-dentable subset L so that on L the weak topology coincides with the norm topology. It follows from known results that the RNP and the KMP are equivalent on subsets of C ( ω ω k ) .

On the structure of the set of higher order spreading models

Bünyamin Sarı, Konstantinos Tyros (2014)

Studia Mathematica

We generalize some results concerning the classical notion of a spreading model to spreading models of order ξ. Among other results, we prove that the set S M ξ w ( X ) of ξ-order spreading models of a Banach space X generated by subordinated weakly null ℱ-sequences endowed with the pre-partial order of domination is a semilattice. Moreover, if S M ξ w ( X ) contains an increasing sequence of length ω then it contains an increasing sequence of length ω₁. Finally, if S M ξ w ( X ) is uncountable, then it contains an antichain of size...

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