A class of maps for the non-separable Banach space renorming theory
by
José Orihuela
Murcia University
Coauthors: A. Moltó (Valencia University, Spain), S. Troyanski (Sofia University, Bulgaria), M. Valdivia (Valencia University, Spain)

The existence of locally uniformly rotund norms on a given non separable Banach space has a considerable impact in his geometry and topology. For instance every discrete family of sets for the norm topology {A_i; i in I} can be decomposed with a countable partition:

Ai = È {Ai, n; n = 1, 2, ...}

in such a way that every one of the families {A_{i, n}; n = 1, 2, ...} for i in I verifies the following " strong discreteness property":

Ai, n Ç conv   {Aj, n, j =/= i, j in I} = \emptyset

that we call "half-space isolated family" because of the Hanh Banach theorem. Consequently every Banach space with a locally uniformly rotund norm has a network for the norm topology which is \sigma -half-space isolated. This condition indeed characterizes the Banach spaces admitting an equivalent locally uniformly rotund norm [6] and it is the beginning of our "topological approach" to renormings, which is strongly connected with the \sigma-fragmentability initiated and developed by J. Jayne, I. Namioka and C. A. Rogers [4, 5] and with the descrptive spaces of R. Hansell [3] . Based on this result we develop a non linear transfer technique for locally uniformly rotund renormings on a given Banach space X, that is we study conditions on a non linear map \Phi from a Banach space X to a Banach space Y with locally uniformly rotund norm in order to transfer this property of the norm to the space X itself. All the known results until now deal with linear maps only and they are far away of the non linear theory which is now developing so fast . We introduce a new class of maps between a topological space (X, \tau) and a metric space (Y, \rho), that goes back to the \sigma -relatively discrete function basis of R. Hansell, [2], and which are related with the subdifferentiability of Lipschitz maps when we consider (X, \tau) to be a Banach space X with \tau the weak topology. With these maps we are able to characterize the good ones \Phi for the transfer technique and we relate them with the topological results we have in [7] too. In C(K) spaces where K is a compact Hausdorff space we have a canonical map to c₀(\Gamma). Indeed we know that K is a subspace of some [0, 1]^\Gamma and the uniform continuity of every x in C(K) allows us to define the oscillation map \Omega(x) in c₀(\Gamma) [1], where

\Omega(x)(\gamma) = sup { |x(s)-x(t)|:s, t in K,  (s-t)|\Gamma\{\gamma}=0 } .

We study when the oscillation map transfers the locally uniformly rotund norm we always have in c₀(\Gamma) and we obtain, for instance, that the Helly space, i.e. is the subspace H of [0, 1]^{[0, 1]} consisting of all nondecreasing functions t :[0, 1] --> [0, 1] endowed with the pointwise topology verifies that C(H) admits a pointwise l.s.c. locally uniformly rotund equivalent norm, a problem posed by M. I. Kadets in the middle of the 70's.

References

[1] R. Engelking, On functions defined in Cartesian products. Fundamenta Math. 59 (1966), 221-231.

[2] R.W. Hansell, Descriptive Topology. Recent Progress in General Topology, North-Holland, Amsterdam, London, NewYork, Tokyo, (1992), 275-315.

[3] R.W. Hansell, Descriptive sets and the topology of nonseparable Banach spaces. Preprint, (1989).

[4] J.E. Jayne, I. Namioka and C.A. Rogers. Topological properties of Banach spaces.Proc. London Math. Soc. 66 (1993) 651-672.

[5] J.E. Jayne, I. Namioka and C.A. Rogers. \sigma- fragmentable Banach spacesMathematika. 39 (1992) 161-188, 197-215.

[6] A. Moltó, J. Orihuela, and S. Troyanski, Locally uniformly rotund renorming and fragmentability. Proc. London Math. Soc., 75, (1997), 619-640.

[7] A. Moltó, J. Orihuela, S. Troyanski and M. Valdivia, On weakly locally uniformly rotund Banach spaces. J. Functional Anal. 163, (1999), 252-271

[8] A. Moltó, J. Orihuela, S. Troyanski and M. Valdivia, Non linear transfer technique. Preprint.

Date received: June 26, 2000