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Isometries and small bound isomorphisms of function spaces. (English) Zbl 0804.46030
Function spaces, Proc. Conf., Edwardsville/IL (USA) 1990, Lect. Notes Pure Appl. Math 136, 241-271 (1992).
[For the entire collection see Zbl 0746.00071.] This paper gives a survey of some results (due to, e.g., the authors, Behrends, Cambern, and many others) connected with the Banach-Stone theorem and with corresponding characterizations of isometries and small bound isomorphisms on various function spaces and function algebras. Here are the contents. Section 3: Banach-Stone theorem, Section 4: (injective, surjective) isometries between subspaces of spaces of continuous functions, Section 5: isometries between Banach function spaces and algebras (5.1 differentiable functions, 5.2 absolutely continuous functions, 5.3 Lipschitz functions, 5.4 general Banach function spaces, 5.5 semisimple commutative Banach algebras), Section 6: isometries of spaces of vector valued functions (6.1 $E$-valued continuous functions, 6.2 injective tensor products of Banach spaces, 6.3 $C(K)$-modules, 6.4 vector valued $\text{weak}\sp*$-continuous functions, 6.5 vector valued analytic functions $[H\sp \infty,H\sp 1]$), Section 7: isomorphisms with small bounds (7.1 spaces of continuous functions, 7.2 injective isomorphisms of spaces of continuous functions, 7.3 spaces of differentiable functions, 7.4 spaces of continuous vector valued functions, 7.5 nonlinear Banach- Stone theorem).

46E15Banach spaces of continuous, differentiable or analytic functions
46J10Banach algebras of continuous functions, function algebras
46J15Banach algebras of differentiable or analytic functions, $H^p$-spaces
46M05Tensor products of topological linear spaces
46B04Isometric theory of Banach spaces
46B25Classical Banach spaces in the general theory of normed spaces
46E40Spaces of vector- and operator-valued functions
46H25Normed modules and Banach modules, topological modules