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Uniqueness of unconditional bases in $c_0$-products

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Casazza P., Kalton N. J.

Studia Mathematica

1999

tom 133

nr 3

275-294

We give counterexamples to a conjecture of Bourgain, Casazza, Lindenstrauss and Tzafriri that if X has a unique unconditional basis (up to permutation) then so does $c_0(X)$. We also give some positive results including a simpler proof that $c_0(ℓ_1)$ has a unique unconditional basis and a proof that $c_0(ℓ_{p_n}^{N_n})$ has a unique unconditional basis when $p_n ￬ 1$, $N_{n+1} ≥ 2N_{n}$ and $(p_n-p_{n+1}) logN_{n}$ remains bounded.

The Maurey extension property for Banach spaces with the Gordon-Lewis property and related structures

100%

Casazza P., Nielsen N.

Studia Mathematica

2003

tom 155

nr 1

1-21

The main result of this paper states that if a Banach space X has the property that every bounded operator from an arbitrary subspace of X into an arbitrary Banach space of cotype 2 extends to a bounded operator on X, then every operator from X to an L₁-space factors through a Hilbert space, or equivalently $B(ℓ_{∞},X*) = Π₂(ℓ_{∞},X*)$. If in addition X has the Gaussian average property, then it is of type 2. This implies that the same conclusion holds if X has the Gordon-Lewis property (in particular X could be a Banach lattice) or if X is isomorphic to a subspace of a Banach lattice of finite cotype, thus solving the Maurey extension problem for these classes of spaces. The paper also contains a detailed study of the property of extending operators with values in $ℓ_{p}$-spaces, 1 ≤ p < ∞.

Ograniczanie wyników

2 Studia Mathematica

2 Casazza P.

1 Kalton N. J.

1 Nielsen N.

1 2003

1 1999

Wyniki wyszukiwania

Uniqueness of unconditional bases in $c_0$-products

The Maurey extension property for Banach spaces with the Gordon-Lewis property and related structures