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Subspaces of $L_{p}$, p > 2, determined by partitions and weights

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Alspach D., Tong S.

Studia Mathematica

2003

tom 159

nr 2

207-227

Many of the known complemented subspaces of $L_{p}$ have realizations as sequence spaces. In this paper a systematic approach to defining these spaces which uses partitions and weights is introduced. This approach gives a unified description of many well known complemented subspaces of $L_{p}$. It is proved that the class of spaces with such norms is stable under (p,2) sums. By introducing the notion of an envelope norm, we obtain a necessary condition for a Banach sequence space with norm given by partitions and weights to be isomorphic to a subspace of $L_{p}$. Using this we define a space Yₙ with norm given by partitions and weights with distance to any subspace of $L_{p}$ growing with n. This allows us to construct an example of a Banach space with norm given by partitions and weights which is not isomorphic to a subspace of $L_{p}$.

Geometry of the Banach spaces C(βℕ × K,X) for compact metric spaces K

100%

Alspach D., Galego E.

Studia Mathematica

2011

tom 207

nr 2

153-180

A classical result of Cembranos and Freniche states that the C(K,X) space contains a complemented copy of c₀ whenever K is an infinite compact Hausdorff space and X is an infinite-dimensional Banach space. This paper takes this result as a starting point and begins a study of conditions under which the spaces C(α), α < ω₁, are quotients of or complemented in C(K,X). In contrast to the c₀ result, we prove that if C(βℕ ×[1,ω],X) contains a complemented copy of $C(ω^{ω})$ then X contains a copy of c₀. Moreover, we show that $C(ω^{ω})$ is not even a quotient of $C(βℕ ×[1,ω],ℓ_{p})$, 1 < p < ∞. We then completely determine the separable C(K) spaces which are isomorphic to a complemented subspace or a quotient of a $C(βℕ ×[1,α],ℓ_{p})$ space for countable ordinals α and 1 ≤ p < ∞. As a consequence, we obtain the isomorphic classification of the $C(βℕ ×K,ℓ_{p})$ spaces for infinite compact metric spaces K and 1 ≤ p < ∞. Indeed, we establish the following more general cancellation law. Suppose that the Banach space X contains no copy of c₀ and K₁ and K₂ are infinite compact metric spaces, then the following statements are equivalent: (1) C(βℕ ×K₁,X) is isomorphic to C(βℕ ×K₂,X). (2) C(K₁) is isomorphic to C(K₂). These results are applied to the isomorphic classification of some spaces of compact operators.

Ograniczanie wyników

2 Studia Mathematica

2 Alspach D.

1 Galego E.

1 Tong S.

1 2011

1 2003

Wyniki wyszukiwania

Subspaces of $L_{p}$, p > 2, determined by partitions and weights

Geometry of the Banach spaces C(βℕ × K,X) for compact metric spaces K