The bidual of a tensor product of Banach spaces. (English) Zbl 1114.46012

Cabello Sánchez, Félix; García, Ricardo

Rev. Mat. Iberoam. 21, No. 3, 843-861 (2005).

Let \(X_1, \dots, X_n\) be Banach spaces and let \(X_1\widehat\otimes \dots \widehat\otimes X_n\) denote their completed \(n\)-fold projective tensor product. The authors consider the natural norm one operator \[ \alpha : X^{\ast\ast}_1\widehat\otimes \dots \widehat\otimes X^{\ast\ast}_n\rightarrow (X_1\widehat\otimes \dots \widehat\otimes X_n)^{\ast\ast} \] defined through the Aron-Berner (or Davie-Gamelin) extension of multilinear operators. The main Theorem is as follows. If the biduals \(X^{\ast\ast}_1,\dots, X^{\ast\ast}_n\) have the bounded approximation property, then \(\alpha\) embeds \(X^{\ast\ast}_1\widehat\otimes \dots \widehat\otimes X^{\ast\ast}_n\) in \((X_1\widehat\otimes \dots \widehat\otimes X_n)^{\ast\ast}\) as a locally complemented subspace. Notice that “locally” cannot be dropped here since by F. C. Sánchez, D. Pérez-Garcia and I. Villanueva [Lond. Math. Soc., II. Ser. 74, No. 2, 512–526 (2006; Zbl 1122.46008)] \(\ell_\infty\widehat\otimes\ell_\infty\) is not complemented in \((c_0\widehat\otimes c_0)^{\ast\ast}\). An immediate consequence is that the continuous \(n\)-linear forms \({\mathcal L}(^nX^{\ast\ast})\) form a complemented subspace of \(({\mathcal L}(^nX))^{\ast\ast}\) whenever \(X^{\ast\ast}\) has the bounded approximation property.
The authors also provide conditions when \(\alpha\) is an isomorphic embedding. In particular, they show relying on a result of [G. Pisier, Math. Ann. 276, 105–136 (1986; Zbl 0619.47016)] that \(\alpha\) embeds \(X^{\ast\ast}\widehat\otimes X^{\ast\ast}\) in \((X\widehat\otimes X)^{\ast\ast}\) isomorphically whenever \(X\) has type 2 and \(X^\ast\) has cotype 2. This applies, for instance, if \(X\) is any \(C^\ast\)-algebra, thus recovering a result from [A. Kumar, A. M. Sinclair, Trans. Am. Math. Soc. 350, No. 5, 2033–2048 (1998; Zbl 0906.46043)]. The paper also presents applications to the Dunford-Pettis property and to holomorphic functions.

Reviewer: Eve Oja (Tartu)

MSC:

46B28	Spaces of operators; tensor products; approximation properties
46G20	Infinite-dimensional holomorphy
46G25	(Spaces of) multilinear mappings, polynomials

Keywords:

projective tensor product; approximation properties; biduals; locally complemented subspaces; symmetric tensor product; holomorphic functions

Citations:

Zbl 0619.47016; Zbl 0906.46043; Zbl 1122.46008

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