Intersection properties of balls in tensor products of some Banach spaces. (English) Zbl 0675.46007

Rao, T. S. S. R. K.; Roy, A. K.; Sundaresan, K.

Math. Scand. 65, No. 1, 103-118 (1989).

Let A be a complex Banach space and n,k integers with \(n>k\geq 3\). We say that A has the almost n.k.I.P. if, for every family \(\{B(a_ j,r_ j)\}^ n_{j=1}\) of n closed balls in A such that any k of them intersect, we have \(\cap^{n}_{j=1}B(a_ j,r_ j+\epsilon)=0\), \(\forall \epsilon >0\). (If we can take \(\epsilon =0\), we say that A has the n.k.I.P.). It is known that A is an \(L^ 1\)-predual, i.e. \(A^*\) is isometric to an \(L^ 1\)-space, if and only if A has the 4.3 I.P.
Introducing the space \(H^ n(A^*)=\{(x_ 1,...,x_ n)\in (A^*)^ n:\sum^{n}_{k=1}x_ k=0\}\) with the norm \(\| (x_ 1,...,x_ n)\| =\sum^{n}_{k=1}\| x_ k\|,\) the following are known to be equivalent:
(1) A has the almost n.k.I.P.
(2) \(A^*\) has the \(R_{n,k}\) property, i.e. if \((x_ 1,...,x_ n)\in H^ n(A^*)\), there exist \((z_{i1},...,z_{in})\in H^ n(A^*)\), (1\(\leq i\leq \left( \begin{matrix} n\\ k\end{matrix} \right))\), such that (i) \((x_ 1,...,x_ n)=\sum_{i}(z_{i1},...,z_{in})\), (ii) \(\| x_ j\| =\sum_{i}\| z_{ij}\| \forall j,\) and each \((z_{i1},...,z_{in})\) has at most k non-zero components for all i.
Inspired by the above results and by earlier work on tensor products of partially ordered linear spaces and tensor products of compact convex sets, and in particular of Choquet simplices, in this paper complex \(L^ 1\)-preduals A are characterised, among Banach spaces, by intersection properties of balls in the injective tensor products of A with other Banach spaces. Specifically, the following result is proved.
Theorem: Let A be a closed subspace of C(Q) separating the points of Q where Q is a compact Hausdorff space. The following statements are equivalent:
(i) A is an \(L^ 1\)-predual.
(ii) If \(n>k\geq 3\) and E is any Banach space with the almost n.k.I.P. then A\({\check \otimes}E\) also has the almost n.k.I.P.
(iii) If k is a fixed integer, (k\(\geq 2)\), and E is a finite-dimensional Banach space with 2k.2k-1.I.P. then A\({\check \otimes}E\) has the almost 2k.2k-1.I.P.
Also proved is a result ‘dual’ to the above theorem and which characterises \(L^ 1\)-spaces by means of the \(R_{n,k}\) property discussed above.
The proofs of the above are non-trivial in as much as several preliminary results of independent interest have to be established on route to the main theorems.

Reviewer: A.K.Roy

MSC:

46B20	Geometry and structure of normed linear spaces
46M05	Tensor products in functional analysis
46E15	Banach spaces of continuous, differentiable or analytic functions

Keywords:

intersection properties of balls; L(sup 1) -predual; tensor products of partially ordered linear spaces; tensor products of compact convex sets; Choquet simplices; injective tensor products