Extreme Points of the Unit Ball in the Dual Space of Some Real Subspaces of Banach Spaces of Lipschitz Functions

Let X be a compact Hausdorff space, τ be a continuous involution on X and C(X,τ) denote the uniformly closed real subalgebra of C(X) consisting of all f∈C(X) for which f∘τ=f̅. Let (X,d) be a compact metric space and let Lip(X,dα) denote the complex Banach space of complex-valued Lipschitz functions of order α on (X,d) under the norm ∥f∥X,pα=max⁡{∥f∥X,pα(f)}, where α∈(0,1]. For α∈(0,1), the closed subalgebra of Lip(X,α) consisting of all f∈Lip(X,dα) for which |f(x)-f(y)|/dα(x,y)→0 as d(x,y)→0, denotes by lip(X,dα). Let τ be a Lipschitz involution on (X,d) and define Lip(X,τ,dα)=Lip(X,dα)∩C(X,τ) for α∈(0,1] and lip(X,τ,dα)=lip(X,dα)∩C(X,τ) for α∈(0,1). In this paper, we give a characterization of extreme points of BA*, where A is a real linear subspace of Lip(X,dα) or lip(X,dα) which contains 1, in particular, Lip(X,τ,dα) or lip(X,τ,dα).