Table of Contents
ISRN Mathematical Analysis
Volume 2012, Article ID 735139, 13 pages
http://dx.doi.org/10.5402/2012/735139
Research Article

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

Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran

Received 20 October 2011; Accepted 16 November 2011

Academic Editor: C. Zhu

Copyright © 2012 Davood Alimohammadi and Hadis Pazandeh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝑋 be a compact Hausdorff space, 𝜏 be a continuous involution on 𝑋 and 𝐢 ( 𝑋 , 𝜏 ) denote the uniformly closed real subalgebra of 𝐢 ( 𝑋 ) consisting of all 𝑓 ∈ 𝐢 ( 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 β€– 𝑓 β€– 𝑋 , 𝑝 𝛼 = m a x { β€– 𝑓 β€– 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 ( π‘₯ ) βˆ’ 𝑓 ( 𝑦 ) | / 𝑑 𝛼 ( π‘₯ , 𝑦 ) β†’ 0 as 𝑑 ( π‘₯ , 𝑦 ) β†’ 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 ) ∩ 𝐢 ( 𝑋 , 𝜏 ) for 𝛼 ∈ ( 0 , 1 ] and l i p ( 𝑋 , 𝜏 , 𝑑 𝛼 ) = l i p ( 𝑋 , 𝑑 𝛼 ) ∩ 𝐢 ( 𝑋 , 𝜏 ) for 𝛼 ∈ ( 0 , 1 ) . In this paper, we give a characterization of extreme points of 𝐡 𝐴 βˆ— , 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 ) .