Renormings of the Dual of James Tree Spaces

We discuss renorming properties of the dual of a James tree space JT . We present examples of weakly Lindelöf determined JT such that JT ∗ admits neither strictly convex nor Kadec renorming and of weakly compactly generated JT such that JT ∗ does not admit Kadec renorming although it is strictly convexifiable. The norm of a Banach space is said to be locally uniformly rotund (LUR) if for every for every x0 with ‖x0‖ = 1 and every ε > 0 there exists δ > 0 such that ‖x− x0‖ < ε whenever ‖ x+x0 2 ‖ > 1 − δ. A lot of research during the last decades has been devoted to understanding which Banach spaces have an equivalent LUR norm, and this is still a rather active line of research. In this note we are concerned with this problem in the case of dual Banach spaces. It is a consequence of a result of Fabian and Godefroy [7] that the dual of every Asplund Banach space (that is, a Banach space such that every separable subspace has a separable dual) admits an equivalent norm which is locally uniformly rotund. It is natural to ask whether, more generally, the dual of every Banach space not containing l1 admits an equivalent LUR norm. We shall give counterexamples to this question by looking at the dual of James tree spaces JT over different trees T . However all these examples are nonseparable, and the problem remains open for the separable case. It was established by Troyanski [18] that a Banach space admits an equivalent LUR norm if and only if it admits an equivalent strictly convex norm and also an equivalent Kadec norm. We recall that a norm is strictly convex if its sphere does not contain any proper segment and it is a Kadec norm if the weak and the norm topologies coincide on its sphere. In Section 1 we shall recall the definition of the spaces JT and the main properties that we shall need. In Section 2 we remark that the space JT ∗ has a LUR renorming whenever JT is separable, so they cannot provide any counterexample for the separable case. We also point out the relation which exists between the renorming properties of JT ∗ and those of C0(T̄ ), the space of continuous functions on the completed tree T̄ vanishing at ∞. Haydon [10] gave satisfactory characterizations of those trees Υ for which C0(Υ) admits LUR, strictly convex or Kadec equivalent norm. We show that if C0(T̄ ) has a LUR (respectively strictly convex) norm then also does JT , and that, on the contrary, if JT ∗ has an equivalent Kadec norm, then so does 2000 Mathematics Subject Classification. 46B03,46B26.