On the fixed-point property of unital uniformly closed subalgebras of \(C(X)\) (Q623521)

A Banach space \(X\) has the fixed point property for nonexpansive mappings if every nonexpansive self-mapping of a nonempty, closed, bounded, convex subset of \(X\) has a fixed point. In [Fixed Point Theory Appl.\ 2010, Article ID 362829 (2010; Zbl 1201.46045)], \textit{W.\ Fupinwong} and \textit{S.\ Dhompongsa} gave sufficient conditions for a unital commutative Banach algebra to fail to have the fixed point property. Their conditions imply that unital commutative \(C^*\)-algebras and the space \(C_{\mathbb{R}}(S)\) of continuous real-valued functions on a compact Hausdorff space \(S\) have the fixed point property if and only if they are finite-dimensional. In the article under review, the authors apply the results of the above-mentioned article to identify some complex and real subalgebras of the space \(C(S)\) of continuous complex-valued functions on a compact Hausdorff space \(S\) that fail to have the fixed point property. For example, the authors prove that, if \(S\) is a compact Hausdorff space and \(A\) is a unital, uniformly closed, complex subalgebra of \(C(S)\) such that the set of peak points for \(A\) contains a limit point of \(S\), then \(A\) fails to have the fixed point property. The authors also consider the fixed point property for a class of unital, uniformly closed, real subalgebras of \(C(S)\) that were introduced by \textit{S. H.\ Kulkarni} and \textit{B. V.\ Limaye} in [Can. J. Math. 33, 181--200 (1981; Zbl 0469.46042)]. If \(\tau\) maps a topological space \(S\) into itself and \(\tau\circ\tau(x)= x\) for all \(x\) in \(S\), then \(C(S,\tau) \overset{\text{def}}{=} \{f\in C(S): \overline{f}\circ \tau = f\}\) and \(C_{\mathbb{R}}(S,\tau)\) denotes the set of real-valued functions in \(C(S,\tau)\). The authors note that, if \(S\) is an infinite compact Hausdorff space, then \(C(S,\tau)\) and \(C_{\mathbb{R}}(S,\tau)\) fail to have the fixed point property.