Reflexivity of the Isometry Group of Some Classical Spaces

We investigate the reflexivity of the isometry group and the automorphism group of some important metric linear spaces and algebras. The paper consists of the following sections: 1. Preliminaries. 2. Sequence spaces. 3. Spaces of measurable functions. 4. Hardy spaces. 5. Banach algebras of holomorphic functions. 6. Fréchet algebras of holomorphic functions. 7. Spaces of continuous functions. Introduction This paper is concerned with the reflexivity of the isometry group and the automorphism group of certain particular but important topological vector spaces and algebras. Although we deal mainly with Banach spaces, we are also interested in other (not necessarily locally convex) metric linear spaces and some Fréchet algebras. Reflexivity problems for subalgebras of the algebra of all bounded linear operators acting on a Hilbert space represent one of the most active research areas in operator theory. The study of similar questions concerning sets of linear transformations on Banach algebras rather than on Hilbert spaces was initiated by Kadison [10] and Larson [14]. In [10], motivated by the study of Hochschild cohomology of operator algebras, the reflexivity of the Lie algebra of all derivations on a von Neumann algebra was treated. In [14, Some concluding remarks (5), p. 298], Larson raised the question of the reflexivity of the automorphism group of Banach algebras. This problem was investigated for several algebras in [1, 19, 20, 21, 23]. The present article is a continuation of that work. We describe the results of the paper as follows. The first section is preliminary. In Section 2 we investigate symmetric spaces. We prove that, with the sole exception of l2, every F -space with a symmetric basis has algebraically reflexive isometry group. Curiously enough, the isometry group of any nonseparable “symmetric” space fails to be algebraically reflexive. Section 3 concentrates on Lebesgue spaces. We show the extreme nonreflexivity of The first author was supported in part by DGICYT project PB97-0377 and HI project 1997-0016. The second author was supported by the Hungarian National Foundation for Scientific Research (OTKA) and by the Ministry of Education (FKFP). 1991 Mathematics Subject Classification 47B49, 46B04, 47B10. 1 2 F. CABELLO SÁNCHEZ AND L. MOLNÁR the isometry group of the spaces Lp(μ) (0 < p < ∞) for homogeneous measures μ, thus obtaining that the only infinite dimensional Lebesgue spaces whose isometry groups are reflexive are the sequence spaces lp for p 6= 2. In contrast, the isometry groups of the Hardy spaces Hp (0 < p < ∞) are topologically reflexive for all p 6= 2. This will be proved in Section 4. Sections 5 and 6 deal with algebras of holomorphic functions. We prove that the isometry group and the automorphism group of the disc algebra are topologically reflexive. The same is true for H(Ω), Ω being any simply connected domain in the plane. Furthermore, we consider the ”unbounded” case: it is shown that the automorphism group of the Fréchet algebra H(Ω) is topologically reflexive if and only if Ω 6= C. Finally, we study algebras of continuous functions. 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. Next, we present another Banach space Y whose isometry group is not algebraically reflexive and yet the isometry group of Y ∗ is algebraically reflexive. 1. Preliminaries Let X be a topological vector space and let B(X) be the algebra of all continuous linear operators on X. Given any subset S ⊂ B(X), define refal S = {T ∈ B(X) : Tx ∈ Sx for all x ∈ X} refto S = {T ∈ B(X) : Tx ∈ Sx for all x ∈ X}, where Sx = {Lx : L ∈ S} and the bar stands for the closure in X. The set S is said to be algebraically reflexive if refal S = S and, similarly, S is called topologically reflexive if refto S = S. Thus, reflexive sets of operators are, in some sense, completely determined by their local actions on the underlying space. Sometimes the operators in refal S are said to belong locally to S. Since there is no clear intrinsic reason to restrict our attention to the locally convex setting when dealing with local surjective isometries, we consider F -spaces, not only Banach spaces. Recall from [12, p. 2] that a ∆-norm on a real or complex vector space X is a non-negative real-valued function on X satisfying (1) ‖x‖ > 0 for all 0 6= x ∈ X. (2) ‖αx‖ ≤ ‖x‖ for all x ∈ X and all α ∈ K with |α| ≤ 1. (3) limα→0 ‖αx‖ = 0 for all x ∈ X. (4) ‖x+y‖ ≤ K(‖x‖+‖y‖) for some constantK independent of x, y ∈ X. REFLEXIVITY OF THE ISOMETRY GROUP OF SOME CLASSICAL SPACES 3 A ∆-norm on X induces a metrizable linear topology for which the sets Un = {x ∈ X : ‖x‖ < 1/n} form a neighbourhood base at the origin and, conversely, every linear metrizable topology comes from a ∆-norm. An F norm is a ∆-norm satisfying (5) ‖x+ y‖ ≤ ‖x‖+ ‖y‖ for all x, y ∈ X. Any F -norm induces a translation-invariant metric on X in the obvious way and every invariant metric compatible with the linear structure is induced by some F -norm. An F -space is a complete F -normed space. Finally, a quasi-norm is a ∆-norm which is homogeneous in the sense that (6) ‖αx‖ = |α|‖x‖ for all x ∈ X,α ∈ K. Observe that (6) implies both (2) and (3) so that quasi-norms can be defined by (1), (6) and (4). A quasi-normed space is a vector space together with some specified quasi-norm. Such a space is locally bounded, that is, it has a bounded neighbourhood of zero. Conversely, every locally bounded topology is induced by a quasi-norm. A quasi-Banach space is a complete quasi-normed space. We denote by Iso(X) the group of all surjective (linear) isometries of the ∆-normed space X. Also, when A is a topological algebra, Aut(A) denotes the group of all continuous automorphisms of A. In accordance with what is written above, we call the elements of refal(Iso(X)) and refal(Aut(A)) local surjective isometries and local automorphisms, respectively. One little problem with ∆-norms is that a ∆-norm need not be continuous with respect to the topology it induces. (The continuity of a norm is a consequence of the triangle inequality.) This has some unpleasant consequences. For instance, the operators in refto(Iso(X)) need not be into isometries. An interesting class of continuous quasi-norms is that of the so-called p-norms (0 < p ≤ 1). These are quasi-norms satisfying ‖x+ y‖ ≤ ‖x‖ + ‖y‖, from which continuity immediately follows. Clearly, if ‖ ·‖ is a p-norm, then ‖ · ‖p is an F -norm with the same isometries. 2. Sequence spaces In this section we study sequence spaces. A basis of an F -space X is a sequence (en) so that every x ∈ X has a unique expansion x = ∑∞ n=1 xnen. A basis (en) is said to be symmetric if