We prove that a biseparating map between spaces B (E), and some other Banach algebras, is automatically continuous and an algebra isomorphism.1991 Mathematics Subject Classification. Primary 47L10; Secondary 46H40, 46B28.
We solve the following three questions concerning surjective linear isometries between spaces of Lipschitz functions Lip( X, E) and Lip(Y , F ), for strictly convex normed spaces E and F and metric spaces X and Y :(i) Characterize those base spaces X and Y for which all isometries are weighted composition maps. (ii) Give a condition independent of base spaces under which all isometries are weighted composition maps. (iii) Provide the general form of an isometry, both when it is a weighted composition map and when it is not.In particular, we prove that requirements of completeness on X and Y are not necessary when E and F are not complete, which is in sharp contrast with results known in the scalar context.
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