Automatic Continuity of Lipschitz Algebras

For a compact metric space (K,d), α ∈ (0,1] and f ∈ C(K), let pα(f) = sup{|f(t) − f(s)|/d(t, s)α: t, s ∈ K}. The set Lipα(K, d) = {f ∈ C(K): pα(f) < ∈} with the norm ||f||α = |f|K + pα(f) is a Banach function algebra under pointwise multiplication. The subset lipα (K, D) = {f ∈ Lipα(K, d) : |f(t) − f (s)|/d(t, s) α → 0 as d(t, s) → 0} is a closed subalgebra of Lipα(K, d). For 0 < α < β, lipα(K, d) ⊇ Lipβ(K, d) ⊇ lipβ(K, d) and so they form a one parameter family of algebras ordered by inclusion. Let Aα = Lipα(K, d) or lipα(K, d). For α, β ∈ (0, 1), the relationship between the ideals of Aα and Aβ is examined, and important inclusions of different such ideals derived. This enables us to establish automatic continuity properties of Lipschitz algebras. A homomorphism v: A → B, B a Banach algebra, is said to be eventually continuous if ∃β ≥α such that v| Aβ is continuous for the ||·||β-norm. First, it is shown that in Lipschitz algebras, when α ∈ (0, 12), the eventual continuity is equivalent to nilpotency of the separating ideal in the range algebra. This is used to prove that if α ∈ (0, 12), and v is eventually continuous, then v is continuous on A2α + γ for all γ ∈ (0, 1 − 2α). It is also shown that in these algebras the prime ideals containing a given prime ideal form a chain. All these results are then used to prove that for α ∈ (0, 12) every epimorphism from Aα is eventually continuous. This research extends work of Bade, Curties, and Laursen, who considered these same questions for Cn([0, 1]).