Let X be a real or complex Banach space with norm ∥·∥· Let G denote the set of all isometric automorphisms on X. Then G is a bounded subgroup of the group of all invertible operators GL(X) in B(X). We shall call G the group of isometries with respect to the norm ∥·∥· A bounded subgroup of GL(X) is said to be maximal if it is not contained in any larger bounded subgroup. The Banach space X has maximal norm if G is maximal. Hilbert spaces have maximal norm. For the (real or complex) spaces c0, lp (1≦p<∞), Lp[0,1] (1≦p<∞), Pelczynski and Rolewicz have shown that the standard norms are maximal ([3], pp. 252–265). In finite dimensional spaces the only maximal groups of isometries are the groups of orthogonal transformations. Given any bounded group H in B(X), X can be renormed equivalently so that each T∈H is an isometry, by ‖x‖1=sup{|Tx‖; T∈H}. Therefore corresponding to every maximal subgroup G there is at least one maximal norm for which G is the group of isometries. In this paper we shall investigate those maximal groups G for which there is only one maximal norm with G as its group of isometries.
Light groups of isometries and polyhedrality of Banach spaces
