The symmetric tensor product of a direct sum of locally convex spaces. (English) Zbl 0931.46005

Ansemil, José M.; Floret, Klaus

Stud. Math. 129, No. 3, 285-295 (1998).

Symmetric tensor products and their relation to the full tensor product are of great importance for the theory of polynomials on Banach spaces. Let \(E\) be any Banach space and let \(\bigotimes^n E\) denote the \(n\)-fold (full) tensor product of \(E\) by itself. For any permutation \(\eta\) in the symmetric group \(S_n\) a mapping \(T_\eta: \bigotimes^n E\to\bigotimes^n E\) is given by \(T_\eta(x_1\otimes\cdots\otimes x_n)= x_{\eta(1)}\otimes\cdots\otimes x_{\eta(n)}\). With this notation, the symmetric \(n\)-fold tensor product of \(E\) is defined to be the subspace \[ \bigotimes^n_s E= \Biggl\{z\in\bigotimes^n E\mid T_\eta z= z\text{ for all }\eta\in S_n\Biggr\} \] of all symmetric tensors. In the paper under review it is shown, that for the symmetric tensor product of a direct sum of two Banach spaces \(F_1\) and \(F_2\) the formula \[ \bigotimes^n_s(F_1\oplus F_2)\cong \bigotimes_{k+\ell= n} \Biggl[\bigotimes^k_s F_1\Biggr]\otimes \Biggl[\bigotimes^\ell_s F_2\Biggr] \] holds true algebraically and topologically for all symmetric tensor topologies \(\tau\) on \(\bigotimes^n E\), i.e. all topologies for which the mappings \(T_\eta\) are continuous for each \(\eta\in S_n\). In the final section of this paper this formula is applied to stable Banach spaces \(E\), i.e. to spaces satisfying \(E\cong E\oplus E\). This yields \[ \bigotimes^n_{\tau,s} E\cong \bigotimes^n_\tau E \] linear topologically for all stable Banach spaces and all symmetric tensor topologies \(\tau\). In the particular case \(\tau= \pi\) such a result was obtained earlier by J. C. Díaz and S. Dineen [Ark. Mat. 36, No. 1, 87-96 (1998)].

Reviewer: H.Junek (Potsdam)

MSC:

46A32	Spaces of linear operators; topological tensor products; approximation properties
46G25	(Spaces of) multilinear mappings, polynomials
46M05	Tensor products in functional analysis
46G20	Infinite-dimensional holomorphy
46A03	General theory of locally convex spaces
15A69	Multilinear algebra, tensor calculus
47H60	Multilinear and polynomial operators

Keywords:

symmetric tensor products; full tensor product; polynomials on Banach spaces