Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 268450, 9 pages
Research Article

On The Fixed-Point Property of Unital Uniformly Closed Subalgebras of 𝐶 ( 𝑋 )

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

Received 25 August 2010; Accepted 24 December 2010

Academic Editor: Lai Jiu Lin

Copyright © 2010 Davood Alimohammadi and Sirous Moradi . 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.


Let 𝑋 be a compact Hausdorff topological space and let 𝐶 ( 𝑋 ) and 𝐶 ( 𝑋 ) denote the complex and real Banach algebras of all continuous complex-valued and continuous real-valued functions on 𝑋 under the uniform norm on 𝑋 , respectively. Recently, Fupinwong and Dhompongsa (2010) obtained a general condition for infinite dimensional unital commutative real and complex Banach algebras to fail the fixed-point property and showed that 𝐶 ( 𝑋 ) and 𝐶 ( 𝑋 ) are examples of such algebras. At the same time Dhompongsa et al. (2011) showed that a complex 𝐶 -algebra 𝐴 has the fixed-point property if and only if 𝐴 is finite dimensional. In this paper we show that some complex and real unital uniformly closed subalgebras of 𝐶 ( 𝑋 ) do not have the fixed-point property by using the results given by them and by applying the concept of peak points for those subalgebras.