Reflexivity of the group of surjective isometries on some Banach spaces

Abstract

To the memory of M. M.In this paper we study the problem of algebraic reflexivity of the isometry group of some important Banach spaces. Because of the previous work in similar topics, our main interest lies in the von Neumann -Schatten p-classes of compact operators. The ideas developed there can be used in £ p -spaces, Banach spaces of continuous functions and spin factors as well. Moreover, we attempt to attract the attention to this problem from general Banach spaces geometry view-point. This study, we believe, would provide nice geometrical results. T e B(X), Tx G £x(Vx e X) => T e £ [T € B(X), Tx e £x(Vx e X) => T e £]holds true. This concept of reflexivity is very useful in the analysis of operator algebras (see [10] and the references therein). Most of the published work is about the reflexivity of derivation algebras and automorphism groups. For results on the algebraic reflexivity of derivation algebras see papers by Bresar, Kadison, Larson, Sourour and Semrl [3,4,8,11]. A theorem, due to Shul'man, on the topological reflexivity of a derivation algebra acting on a C*-algebra can be found in [17]. For results on algebraic reflexivity of the automorphism group see papers by Bresar and Semrl [4] and [5], In a recent article [13] of the first author it was proved that the automorphism group of

2-Local Isometries of Some Operator Algebras

“…Finally, we feel that, just as in the case of linear (1-)local isometries (see [8]), it would be interesting to investigate the 2-local isometries of other or more-general Banach spaces.…”

“…In a series of papers (see [1,[6][7][8] and the references cited therein) we investigated the automorphism groups and the isometry groups of operator algebras from the point of view of how they are determined by their local actions. Our investigations were motivated by the paper by Kadison [3] on local derivations and by a problem of Larson in [5] initiating the study of local automorphisms of Banach algebras.…”

Reflexivity of the isometry group of some classical spaces

“…It is natural to ask if the same occurs with any equivalent norm (or renorming, in short). Although in [22] an affirmative answer was conjectured, the following result shows that the answer is strongly negative.…”

“…Therefore, Iso(H) cannot be reflexive unless H is finite dimensional. In fact, the isometry group of any infinite dimensional complex Hilbert space is algebraically nonreflexive not only with respect to the original Hilbert space norm, but also with respect to the so-called spin norms [22,Theorem 3.7]. It is natural to ask if the same occurs with any equivalent norm (or renorming, in short).…”

“…We solve some problems posed in [22] by presenting Banach spaces whose isometry groups are either trivial or very large. We give an example of a compact Hausdorff space K such that the isometry group and the automorphism group of the Banach algebra C(K) both fail to be reflexive, thus verifying a conjecture formulated in [22], where it was proved that the isometry group and the automorphism group of C(K) are algebraically reflexive in case K is first countable. Furthermore, we exhibit a Banach space X with the property that its isometry group is topologically reflexive but the isometry group of its dual space X * is even algebraically nonreflexive.…”