Banach Spaces Having the Radon-nikodỳm Property and Numerical Index 1

Let X be a Banach space with the Radon-Nikodỳm property. Then, the following are equivalent. (i) X has numerical index 1. (ii) |x∗∗(x∗)| = 1 for all x∗ ∈ ex(BX∗ ) and x∗∗ ∈ ex(BX∗∗ ). (iii) X is an almost-CL-space. (iv) There are a compact Hausdorff space K and a linear isometry J : X → C(K) such that |x∗∗(J∗δs)| = 1 for all s ∈ K and x∗∗ ∈ ex(BX∗∗ ). If X is a real space, the above conditions are equivalent to being semi-nicely embedded in some space C(K). The numerical index of a Banach space is a constant relating the norm and the numerical radius of operators on the space. Let us present the relevant definitions. For a Banach space X , we write BX for the closed unit ball and SX for the unit sphere. We denote by X∗ the dual space and by L(X) the Banach algebra of all bounded linear operators on X . For such an operator T , the numerical radius of T is v(T ) = sup{|x∗(Tx)| : x∗ ∈ SX∗ , x ∈ SX , x∗(x) = 1}. The numerical index of the space X is then given by n(X) = max{k ≥ 0 : k ‖T ‖ ≤ v(T ) ∀T ∈ L(X)}. We refer the reader to the books [3, 4] and to the expository paper [13] for general information and background. Recent results can be found in [7, 12, 14, 15]. Let us mention here some facts concerning the numerical index which will be relevant to our discussion. First, one has v(T ∗) = v(T ) for every T ∈ L(X), where T ∗ is the adjoint operator of T (see [3, §9]), and it clearly follows that n(X∗) ≤ n(X). The question whether this is actually an equality seems to be open. Second, it is a classical result that Land M -spaces have numerical index 1 [6]. The aim of this paper is to characterize Banach spaces with numerical index 1 among those having the Radon-Nikodỳm property (RNP for short; see [5] for background). To this end, we prove that some general sufficient conditions for a Banach space to have numerical index 1 are also necessary if the space has the RNP. Received by the editors November 20, 2001. 2000 Mathematics Subject Classification. Primary 46B20, 47A12. This research was partially supported by Spanish MCYT projects no. BFM2000-1467 and