Isometries on extremely non-complex Banach spaces

Abstract

Given a separable Banach space $E$, we construct an extremely non-complex Banach space (i.e. a space satisfying that $\Vert Id+T^2\Vert =1+\Vert T^2\Vert$ for every bounded linear operator $T$ on it) whose dual contains $E^{\ast }$ as an $L$-summand. We also study surjective isometries on extremely non-complex Banach spaces and construct an example of a real Banach space whose group of surjective isometries reduces to $\pm Id$, but the group of surjective isometries of its dual contains the group of isometries of a separable infinite-dimensional Hilbert space as a subgroup.

18 pages, revised version, to appear in J. Inst. Math. Jussieu

References in corpus (2)

• The group of isometries of a Banach space and duality
• Extremely non-complex C(K) spaces