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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 $C\left(K\right)$-modules, 6.4 vector valued ${\text{weak}}^{*}$-continuous functions, 6.5 vector valued analytic functions $\left[{H}^{\infty },{H}^{1}\right]$), 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).

##### MSC:
 46E15 Banach spaces of continuous, differentiable or analytic functions 46J10 Banach algebras of continuous functions, function algebras 46J15 Banach algebras of differentiable or analytic functions, ${H}^{p}$-spaces 46M05 Tensor products of topological linear spaces 46B04 Isometric theory of Banach spaces 46B25 Classical Banach spaces in the general theory of normed spaces 46E40 Spaces of vector- and operator-valued functions 46H25 Normed modules and Banach modules, topological modules