Discontinuous Maps from Lipschitz Algebras

✍ Scribed by Branka Pavlovic

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 340 KB
• Volume: 155
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1997.3232

✦ Synopsis

For an infinite compact metric space (X, d), : # (0, 1) and f # C(X), let p

Both Lip : (X, d ) and lip : (X, d ) are called Lipschitz algebras. We solve the problem of existence of homomorphisms and derivations from lip : (X, d ) which are discontinuous on every dense subalgebra. In order to achieve that, we use the existence of x # X and a linear functional * from m : (x)=[ f # lip : (X, d): f(x)=0] which is discontinuous on every dense subalgebra, but satisfies |*( fg)| p : ( f ) p : (g) for all f, g # m : (x). We present two constructions of such functionals, one for the general case of lip : (X, d ), and another one for the special case lip : ([0, 1]). We relate the present results to the results concerning the eventual continuity of homomorphisms from Lipschitz algebras. We conclude that for any ; # [:, min[2:, 1]) there exist homomorphisms and derivations discontinuous on lip $ (X, d