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.

Abstract

Let be a compact Hausdorff topological space and let and denote the complex and real Banach algebras of all continuous complex-valued and continuous real-valued functions on under the uniform norm on , 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 and are examples of such algebras. At the same time Dhompongsa et al. (2011) showed that a complex -algebra has the fixed-point property if and only if is finite dimensional. In this paper we show that some complex and real unital uniformly closed subalgebras of 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.