Near-rings of continuous functions on compact abelian groups (Q687652)

\(N(G)\) denotes the nearring of all continuous selfmaps of the Hausdorff topological group \(G\) where addition is pointwise and multiplication is composition. The intersection of all nonzero ideals of \(N(G)\) is referred to as the heart of \(N(G)\) and is denoted by \(H(G)\). The symbol \([G,G]\) denotes the nearring of all homotopy classes of continuous selfmaps of \(G\) where the binary operations on \([G,G]\) are those which are induced by the binary operations on \(N(G)\). In the main result of the paper, the author shows that if \(G\) is a compact abelian group with nontrivial connected components, then \(H(G)\) consists of all those functions in \(N(G)\) which are homotopic to the constant function on \(G\) which carries everything to zero. Moreover, the nearrings \(N(G)/H(G)\) and \([G,G]\) are isomorphic. Not only is the theorem interesting, but the manner in which it is proved is interesting as well. Results and techniques developed by \textit{K. H. Hofmann} [``Introduction to the theory of compact groups'', Part I (Tulane Univ. , Dept. Math., 1968; Zbl 0229.22006)] play an important role. To be somewhat more specific, denote by \(L(G)\) the group of all continuous homomorphisms from \(R\) into a compact abelian group \(G\) and endow \(L(G)\) with the compact open topology. Then \(L(G)\) is a topological vector space and the mapping exp from \(L(G)\) to \(G\) defined by \(\text{exp}(\alpha) = \alpha(1)\) is a continuous homomorphism. The author then investigates the ideals of the sandwich nearring \(N(G,L(G),\text{exp})\) of all continuous maps from \(G\) to \(L(G)\) where addition is pointwise and the product \(fg\) of two functions \(f,g \in N(G,L(G),\text{exp})\) is defined by \(fg = f \circ \text{exp} \circ g\). The results obtained are then used in verifying the main theorem.