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 = 𝑓 . Let ( 𝑋 , 𝑑 ) be a compact metric space and let L i p ( 𝑋 , 𝑑 𝛼 ) denote the complex Banach space of complex-valued Lipschitz functions of order 𝛼 on ( 𝑋 , 𝑑 ) under the norm ‖ 𝑓 ‖ 𝑋 , 𝑝 𝛼 = m a x { ‖ 𝑓 ‖ 𝑋 , 𝑝 𝛼 ( 𝑓 ) } , where 𝛼 ∈ ( 0 , 1 ] . For 𝛼 ∈ ( 0 , 1 ) , the closed subalgebra of L i p ( 𝑋 , 𝛼 ) consisting of all 𝑓 ∈ L i p ( 𝑋 , 𝑑 𝛼 ) for which | 𝑓 ( 𝑥 ) − 𝑓 ( 𝑦 ) | / 𝑑 𝛼 ( 𝑥 , 𝑦 ) → 0 as 𝑑 ( 𝑥 , 𝑦 ) → 0 , denotes by l i p ( 𝑋 , 𝑑 𝛼 ) . Let 𝜏 be a Lipschitz involution on ( 𝑋 , 𝑑 ) and define L i p ( 𝑋 , 𝜏 , 𝑑 𝛼 ) = L i p ( 𝑋 , 𝑑 𝛼 ) ∩ 𝐶 ( 𝑋 , 𝜏 ) for 𝛼 ∈ ( 0 , 1 ] and l i p ( 𝑋 , 𝜏 , 𝑑 𝛼 ) = l i p ( 𝑋 , 𝑑 𝛼 ) ∩ 𝐶 ( 𝑋 , 𝜏 ) for 𝛼 ∈ ( 0 , 1 ) . In this paper, we give a characterization of extreme points of 𝐵 𝐴 ∗ , where 𝐴 is a real linear subspace of L i p ( 𝑋 , 𝑑 𝛼 ) or l i p ( 𝑋 , 𝑑 𝛼 ) which contains 1, in particular, L i p ( 𝑋 , 𝜏 , 𝑑 𝛼 ) or l i p ( 𝑋 , 𝜏 , 𝑑 𝛼 ) .
