Automatic Continuity of Lipschitz Algebras

Abstract

For a compact metric space ( K, d), α ∈ (0,1] and f ∈ C( K), let p α( f) = sup{| f( t) − f( s)|/ d( t, s) α: t, s ∈ K}. The set Lip α( K, d) = { f ∈ C( K): p α( f) < ∈} with the norm || f|| α = | f| K + p α( f) is a Banach function algebra under pointwise multiplication. The subset lip α ( K, D) = { f ∈ Lip α( K, d) : | f( t) − f ( s)|/ d( t, s) α → 0 as d( t, s) → 0} is a closed subalgebra of Lip α( K, d). For 0 < α < β, lip α( K, d) ⊇ Lip β( K, d) ⊇ lip β( K, d) and so they form a one parameter family of algebras ordered by inclusion. Let A α = Lip α( K, d) or lip α( K, d). For α, β ∈ (0, 1), the relationship between the ideals of A α and A β 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| A β is continuous for the ||·|| β-norm. First, it is shown that in Lipschitz algebras, when α ∈ (0, 1 2 ), the eventual continuity is equivalent to nilpotency of the separating ideal in the range algebra. This is used to prove that if α ∈ (0, 1 2 ), and v is eventually continuous, then v is continuous on A 2α + 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, 1 2 ) every epimorphism from Aα is eventually continuous. This research extends work of Bade, Curties, and Laursen, who considered these same questions for C n ([0, 1]).