The group of isometries of a Banach space and duality

Abstract

We construct an example of a real Banach space whose group of surjective isometries has no uniformly continuous one-parameter semigroups, but the group of surjective isometries of its dual contains infinitely many of them. Other examples concerning numerical index, hermitian operators and dissipative operators are also shown.Comment: To appear in J. Funct. Ana

Banach space characterizations of unitaries: A survey

“…Indeed, as a consequence of [49,Theorem 3.3] (with E equal to the two-dimensional Euclidean real space), there is a real Banach space X (namely, the space X(E) in that theorem) such that the identity mapping 1 on X is a geometrically unitary element of L( X) (with n(L( X), 1) = 1), whereas the identity mapping on X * is not even a vertex of B L( X * ) .…”

“…(Theorem 2.1), as well as some results, both in the theory of Banach algebras [8,13,14,18,43,54,57] and in the one of Banach spaces [4,34,[49][50][51], originated in that characterization. It is worth mentioning that the Bandyopadhyay-Jarosz-Rao paper [4] is motivated by the recent Akemann-Weaver rediscovery [2] of the Bohnenblust-Karlin characterization, and that, in its turn, some results in [4] become rediscoveries of previous ones in [14,51].…”

Isometries on extremely non-complex Banach spaces

“…The next result describes the dual of a C E (K L) space for an arbitrary E ⊆ C(L). It is worth mentioning that its proof is an extension of that appearing in [21,Theorem 3.3].…”

Polynomial Numerical Index for Some Complex Vector-Valued Function Spaces

“…Nevertheless, as it was commented in the introduction, It is possible to prove that the same example works for the polynomial case. Anyhow, we prefer to give an example using the approach given in the very recent paper [23], which will allow us to present a space showing that the behavior of the polynomial numerical index with respect to the biduality is the worst possible. We will use Theorem 3.3 of [23], actually part of its proof.…”