Complemented copies of c0(τ) in tensor
products of Lp[0,1]

Cortes, Vinícius Morelli; Galego, Elói Medina; Samuel, Christian

doi:10.2140/pjm.2019.301.67

Cited by 4 publications

(3 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since the submission of this paper, the authors have shown that Problem has a positive solution provided that τ has uncountable cofinality. This result will appear in .…”

mentioning

confidence: 65%

When is complemented in tensor products of ?

Cortes

Galego

Samuel

2018

Mathematische Nachrichten

Self Cite

View full text Add to dashboard Cite

Let be a Banach space, let be an infinite set, let be an infinite cardinal and let ∈ [1, ∞). In contrast to a classical 0 result due independently to Cembranos andFreniche, we prove that if the cofinality of is greater than the cardinality of , then the injective tensor product ( )⊗ contains a complemented copy of 0 ( ) if and only if does. This result is optimal for every regular cardinal . On the other hand, we provide a generalization of a 0 result of Oja by proving that if is an infinite cardinal, then the projective tensor product ( )⊗ contains a complemented copy of 0 ( ) if and only if does. These results are obtained via useful descriptions of tensor products as convenient generalized sequence spaces. K E Y W O R D S 0 (Γ) spaces, complemented subspaces, injective tensor product, ( ) spaces, projective tensor product M S C ( 2 0 1 0 ) Primary: 46B03, 46B15; Secondary: 46E30, 46E40 ∈ ‖ ‖ < ∞ } ,

show abstract

“…Since the submission of this paper, the authors have shown that Problem has a positive solution provided that τ has uncountable cofinality. This result will appear in .…”

mentioning

confidence: 65%

When is complemented in tensor products of ?

Cortes

Galego

Samuel

2018

Mathematische Nachrichten

Self Cite

View full text Add to dashboard Cite

show abstract

“…The first part of our investigation concerns C(K, X) spaces and goes back to the classical and celebrated Cembranos-Freniche theorem [7] which states that C(K, X) has a complemented subspace isomorphic to c 0 whenever K is an infinite Hausdorff compactum and X is an infinite dimensional Banach space. The Cembranos-Freniche theorem in the past decades has influenced many lines of research, for recent examples [4] [9] [10], and was extended in many directions [27] [26] [14]. Galego and Hagler in [14], among other results, isolated conditions on K and X implying that C(K, X) will have a complemented subspace isomorphic to c 0 (Γ ), where Γ is an infinite set, not necessarily countable.…”

Section: Introductionmentioning

confidence: 99%

Complementations in $C(K,X)$ and $\ell_\infty(X)$

Candido¹

2021

Preprint

View full text Add to dashboard Cite

We investigate the geometry of C(K, X) and ℓ∞(X) spaces through complemented subspaces of the form i∈Γ Xi c 0 . Concerning the geometry of C(K, X) spaces we extend some results of D. Alspach and E. M. Galego from [1]. On ℓ∞-sums of Banach spaces we prove that if ℓ∞(X) has a complemented subspace isomorphic to c0(Y ), then, for some n ∈ N, X n has a subspace isomorphic to c0(Y ). We further prove the following:(1) If C(K) ∼ c0(C(K)) and C(L) ∼ c0(C(L)) and ℓ∞(C(K)) ∼ ℓ∞(C(L)), then K and L have the same cardinality. (2) if K1 and K2 are infinite metric compacta, then ℓ∞(C(K1)) ∼ ℓ∞(C(K2)) if and only if C(K1)is isomorphic to C(K2).

show abstract

“…The first part concerns C(K, X) spaces and goes back to the celebrated Cembranos-Freniche theorem [7] which states that C(K, X) has a complemented subspace isomorphic to c 0 whenever K is an infinite Hausdorff compact space and X is an infinite-dimensional Banach space. The Cembranos-Freniche theorem has influenced several lines of research in the past decades [6,9,10,18] and was extended in many directions [14,27,28]. Galego and Hagler [14], among other results, isolated conditions on K and X yielding a complemented subspace in C(K, X) isomorphic to c 0 (Γ ), where Γ is an infinite set, not necessarily countable.…”

mentioning

confidence: 99%

Complementations in $C(K,X)$ and $\ell _\infty (X)$

Candido¹

2023

Colloq. Math.

View full text Add to dashboard Cite

We investigate the geometry of C(K, X) and ℓ∞(X) spaces through complemented subspaces of the form ( i∈Γ Xi)c 0 . For Banach spaces X and Y , we prove that if ℓ∞(X) has a complemented subspace isomorphic to c0(Y ), then, for some n ∈ N, X n has a subspace isomorphic to c0(Y ). If K and L are Hausdorff compact spaces and X and Y are Banach spaces having no subspace isomorphic to c0 we further prove the following:(1) If C(K) ∼ c0(C(K)) and C(L) ∼ c0(C(L)) and ℓ∞(C(K, X)) ∼ ℓ∞(C(L, Y )), then K and L have the same cardinality. (2) If K and L are infinite and metrizable and ℓ∞(C(K, X)) ∼ ℓ∞(C(L, Y )), then C(K) is isomorphic to C(L).

show abstract

Complemented copies of c0(τ) in tensor products of Lp[0,1]

Cited by 4 publications

References 26 publications

When is complemented in tensor products of ?

When is complemented in tensor products of ?

Complementations in $C(K,X)$ and $\ell_\infty(X)$

Complementations in $C(K,X)$ and $\ell _\infty (X)$

Contact Info

Product

Resources

About