Abstract
Here we describe our results and their background: terminology (mostly standard) is denned in Section 2. Throughout, F is a separable Banach space, 1 ≤ p < ∞ and Lp(F) is the space of measurable functions [0,1] → F with P-integrable norms. Given a ‘nice’ property P for Banach spaces, we may formulate the conjecture: Lp(F) satisfies P if and only if both F and Lp (= LP(ℝ)) satisfy P. This conjecture is known to be true for various specific properties, for example the Radon–Nikodym property ((4), section 5·4); reflexivity ((4), corollary 4·1·2); super-refiexivity ((12), proposition 1·2); B-convexity ((14), p. 200); and the properties of not containing copies of c0 (6) and l1 (13). The object of this paper is to demonstrate that the conjecture is false for the property of having an unconditional basis – this answers a question in (4).
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