When is c0(τ) complemented in tensor products of ℓp(I)?

Vinícius Morelli Cortes, Elói Medina Galego and Christian Samuel

Mathematische Nachrichten, 2019, vol. 292, issue 5, 1089-1105

Abstract: Let X be a Banach space, let I be an infinite set, let τ be an infinite cardinal and let p∈[1,∞). In contrast to a classical c0 result due independently to Cembranos and Freniche, we prove that if the cofinality of τ is greater than the cardinality of I, then the injective tensor product ℓp(I)⊗̂εX contains a complemented copy of c0(τ) if and only if X does. This result is optimal for every regular cardinal τ. On the other hand, we provide a generalization of a c0 result of Oja by proving that if τ is an infinite cardinal, then the projective tensor product ℓp(I)⊗̂πX contains a complemented copy of c0(τ) if and only if X does. These results are obtained via useful descriptions of tensor products as convenient generalized sequence spaces.

Date: 2019
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