Discontinuous Maps from Lipschitz Algebras

Abstract

For an infinite compact metric space (X, d),α∈(0, 1) andf∈C(X), letpα(f)=sup{|f(t)−f(s)|/d(t, s)α:t, s∈X, t≠s}. The set Lipα(X, d)={f∈C(X):pα(f)<∞} with the norm ‖f‖α=|f|X+pα(f) is a Banach function algebra under pointwise multiplication. The subset lipα(X, d)={f∈Lipα(X, d):|f(t)−f(s)|/d(t, s)α→0 asd(t, s)→0} is a closed subalgebra of Lipα(X, d). Both Lipα(X, d) and lipα(X, d) are called Lipschitz algebras. We solve the problem of existence of homomorphisms and derivations from lipα(X, d) which are discontinuous on every dense subalgebra. In order to achieve that, we use the existence ofx∈Xand a linear functionalλfrommα(x)={f∈lipα(X, d):f(x)=0} which is discontinuous on every dense subalgebra, but satisfies |λ(fg)|⩽pα(f)pα(g) for allf, g∈mα(x). We present two constructions of such functionals, one for the general case of lipα(X, d), and another one for the special case lipα([0, 1]). We relate the present results to the results concerning the eventual continuity of homomorphisms from Lipschitz algebras. We conclude that for anyβ∈[α, min{2α, 1}) there exist homomorphisms and derivations discontinuous on lipδ(X, d) in ‖·‖δnorm for allδ∈[α, β]. In our constructions, all maps are continuous forδ>β.