The group of isometries and the structure of a finite-dimensional Banach space

Abstract

Let X be a finite-dimensional complex Banach space. The set G of all isometries on X is a compact Lie group. Let G 0 be the identity component of G. Further, denote by H(X) the set of all Hermitian operators on X. It is shown that the space H(X)+i H(X) is an algebra if and only if there exists a system of r orthogonal non-zero Hermitian idempotents on X, where r= rankG 0. An easy consequence of this theorem is a result of H. Schneider and R.E.L. Turner on matrices Hermitian for an absolute norm.