Intersection properties of balls in tensor products of some Banach spaces. II. (English) Zbl 0706.46019

Rao, T. S. S. R. K.

Indian J. Pure Appl. Math. 21, No. 3, 275-284 (1990).

Summary: We study an intersection property of balls called the finite intersection property, for injective and projective tensor products of Banach spaces. It turns out that for a Banach space E containing an isometric copy of \(c_ 0\), E\({\check \otimes}F\) fails this property when F is infinite dimensional. A precise positive answer can be obtained when E is a space of continuous functions. Similarly this intersection property will not be in general preserved by projective tensor products. By establishing some general theorems about proper M-ideals we conclude that several of the classical compact operator spaces fail this property.
[For part I see Math. Scand. 65, No.1, 103-108 (1989; Zbl 0675.46007).]

MSC:

46B20	Geometry and structure of normed linear spaces
46B28	Spaces of operators; tensor products; approximation properties
46M05	Tensor products in functional analysis
46E15	Banach spaces of continuous, differentiable or analytic functions

Keywords:

intersection property of balls; finite intersection property; injective and projective tensor products of Banach spaces; proper M-ideals; compact operator spaces

Citations:

Zbl 0675.46007