Light groups of isometries and polyhedrality of Banach spaces

Abstract

Megrelishvili defines light groups of isomorphisms of a Banach space as the groups on which the weak and strong operator topologies coincide and proves that every bounded group of isomorphisms of Banach spaces with the point of continuity property (PCP) is light. We investigate this concept for isomorphism groups G of classical Banach spaces X without the PCP, especially isometry groups, and relate it to the existence of G-invariant LUR or strictly convex renormings of X. We give an example of a Banach space X and an infinite countable group of isomorphisms G GL(X) which is SOT-discrete but such that X does not admit a distinguished point for G, providing a negative answer to a question of Ferenczi and Rosendal. We also prove that every combinatorial Banach space is (V)polyhedral. In particular, the Schreier spaces of countable order provide new solutions to a problem proposed by Lindenstrauss concerning the existence of an infinite-dimensional Banach space whose unit ball is the closed convex hull of its extreme points.

Isometries of combinatorial Banach spaces

“…where the three vectors are disjointly supported (possibly y * i or z * i is 0), and y * i and z * i only have coordinates ±1 on their support. From the fact that for i = j, T (e * p i ± e * p j ) must be an extreme point and therefore does not have ± 1 2 coordinates, we deduce that the support of y * i is some finite set C independent of i and that z * i is disjointly supported from k, from y * i and from all other z * j . Since C is finite we find i = j such that y * i = y * j , and we compute The following example shows that a bounded operator that sends extreme points to extreme points and vectors of disjoint support to vectors of disjoint support is not necessarily given by a permutation of the basis.…”

“…In a similar manner, reversing the roles of n and m we obtain that α m k = ± 1 2 as well. Plugging these values in (1), and taking into account that all θ j α j k are non-negative, it follows that α j k = 0 for all j ∈ F \ {n, m}.…”

“…Let us observe that the dual of any combinatorial space has the CRSP (see [2], [1]), which means that any element of the unit ball is a sequentially convex combination of extreme points. So in particular the convex hull of extreme points is norm dense in the unit ball and this implies that any map between duals of combinatorial spaces sending extreme points to extreme points must be a contraction.…”