When does C(K,X) contain a complemented copy of c0(Γ) iff X does?

Cortes, Vinícius Morelli; Galego, Elói Medina

doi:10.1016/j.bulsci.2020.102839

Cited by 2 publications

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“…Note also that (1.2) follows from definitions. On the other hand, proof of (1.3) and (1.4) uses the following result whose proof can be found in [6]:…”

Section: Introductionmentioning

confidence: 99%

Embedding c0(Γ) and ℓ∞(Γ) in Banach spaces of homogeneous polynomials

Rincón,

Pérez,

Reyes

2023

Preprint

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We study Pw∗ (nX∗, Y), the Banach space of all n−homogeneous polynomials which are w∗ −w−continuous, endowed with the sup norm. The following facts are proved: (1) c0(Γ) embeds in Pw∗ (nX∗, Y ) iff either ℓ∞(Γ) embeds in Pw∗ (nX∗, Y ) or c0(Γ) embeds in the sum X ⊕ Y. (2) If c0(Γ) is isomorphic to a complemented subspace of Pw∗ (nX∗, Y ), then c0(Γ) embeds in X ⊕ Y . We also extend some linear techniques in order to embed complementably c0(Γ) in either Pw∗ (nX∗, Y ) or P(nX, Y), the Banach space of all n−homogeneous polynomials from X to Y , endowed with the sup norm. Finally, if Pw∗K(nX∗, Y) is the closed subspace of all compact polynomials in Pw∗ (nX∗, Y), we prove that ℓ∞ embeds in Pw∗K (nX∗, Y) iff ℓ∞embeds in either X or Y . As a consequence, we prove that if c0 embeds in Pw∗K(nX∗, Y), then Pw∗K(nX∗, Y ) = nX∗ (nX∗, Y) iff only one of the following statements is true: (1) c0 embeds in Y and X has the Schur property, (2) c0 embeds in X and Y has the Schur property. 2020 Mathematics Subject Classification. 46B03, 46B25, 46E40, 46G25.

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“…Note also that (1.2) follows from definitions. On the other hand, proof of (1.3) and (1.4) uses the following result whose proof can be found in [6]:…”

Section: Introductionmentioning

confidence: 99%