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

Davood Alimohammadi,Sirous Moradi

doi:10.1155/2010/268450

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. (2010) 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.

Highlights

Introduction and PreliminariesWe let C, R, N {1, 2, 3, . . .}, T {z ∈ C : |z| 1}, D {z ∈ C : |z| < 1}, D {z ∈ C : |z| ≤ 1} denote the fields of complex, real numbers, the set of natural numbers, the unit circle, the open unit disc, and the closed unit disc, respectively
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
We denote by CF X the unital commutative Banach algebra over F of continuous functions from X to F with pointwise addition, scalar multiplication, and product with the uniform norm f X sup f x : x ∈ X f ∈ CF X

Summary

Introduction and Preliminaries

We let C, R, N {1, 2, 3, . . .}, T {z ∈ C : |z| 1}, D {z ∈ C : |z| < 1}, D {z ∈ C : |z| ≤ 1} denote the fields of complex, real numbers, the set of natural numbers, the unit circle, the open unit disc, and the closed unit disc, respectively. Let τ be a topological involution on a compact Hausdorff topological space X and let A be a unital uniformly closed real subspace of C X, τ. Let τ be a topological involution on a compact Hausdorff topological space X and A be a unital uniformly closed real subspace of C X, τ. Let A be an infinite dimensional unital commutative complex Banach algebra satisfying each of the following:. We give a general condition for some infinite dimensional unital uniformly closed subalgebras of C X to fail the fixed-point property by applying Theorems 1.4 and 1.6. By using the concept of τ-peak points for unital uniformly closed real subalgebras of C X, τ , we show that some of these algebras do not have the fixed-point property

FPP of Complex Subalgebras of C X

FPP of Real Subalgebras of C X