Reflexivity of the isometry group of some classical spaces. (English) Zbl 1050.47028

The paper under review is a continuation of investigations initiated by Kadison and Larson who studied the reflexivity of the Lie algebra of all derivations on a von Neumann algebra, motivated by the study of the Hochschild cohomology of operator algebras.
The authors investigate algebraic reflexivity and topological reflexivity of Iso\((X)\), the group of all isometric surjections of a \(\Delta\)-normed vector space \(X\), and of Aut\(({\mathcal{A}})\), the group of all continuous automorphisms of a topological algebra \(\mathcal{A}\). To this end, they concentrate their related investigation on (quasi-)Banach spaces and Banach algebras which play a fundamental role in harmonic analysis such as the spaces \(L_p(\mu)\) and the Hardy space \(H^p\), where \(0 <p < \infty\); \(H(\Omega)\), the Banach algebra of all holomorphic functions on a domain \(\Omega\) in \({\mathbb{C}}\); \({H^{\infty}}(\Omega)\), the Banach algebra of all bounded functions in \(H(\Omega)\); and A\(({\overline{U}})\), the Banach algebra of all continuous functions on the closure of the open unit disc \(U:= \{z \in {\mathbb{C}} \mid | z | < 1 \}\) which are holomorphic on \(U\).
In particular, they show that for any infinite-dimensional \(L_p\)-space \(X\), Iso\((X)\) is algebraically reflexive if and only if \(X\) is (isometrically isomorphic to) the sequence space \(l_p\), where \(p \not= 2\) (Theorem 3). Moreover, since every Banach space \(X\) has an equivalent norm so that Iso\((X)\) consists of trivial isometries only, the authors’ considerations show the unpleasant fact that the validity of topological reflexivity of Iso\((X)\) strongly depends on the choice of the equivalent norm on the Banach space \(X\) (Theorem 2).


47B38 Linear operators on function spaces (general)
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
47B48 Linear operators on Banach algebras
46B04 Isometric theory of Banach spaces
46B20 Geometry and structure of normed linear spaces
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