On The Fixed-Point Property of Unital Uniformly Closed Subalgebras of

Abstract

Let X be a compact Hausdorff topological space and let C X and C R 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 R 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.

The Fixed Point Property of a Banach Algebra Generated by an Element with Infinite Spectrum

“…In 2010, D. Alimohammadi and S. Moradi [6] used the above result to obtain sufficient conditions to show that some unital uniformly closed subalgebras of (Ω), where Ω is a compact space, do not have the fixed point property.…”