Compact homomorphisms on algebras of continuous functions (Q1913565)

The purpose of this note is to study compact and weakly compact homomorphisms between algebras of continuous functions. For a completely regular Hausdorff space \(S\), we denote by \(C(S)\) the algebra of all complex-valued continuous functions on \(S\) endowed with its compact-open topology. \textit{M. Lindström} and \textit{J. Llavona} [J. Math. Anal. Appl. 166, No. 2, 325-330 (1992; Zbl 0802.46065)] gave characterizations of compact and weakly compact homomorphisms from \(C(S)\) to \(C(T)\), where \(T\) and \(S\) are completely regular Hausdorff spaces. Let \(A\) and \(B\) be closed subalgebras of \(C(S)\) and \(C(T)\) respectively. Here, we study compact and weakly compact homomorphisms \(\varphi\) from \(A\) to \(B\). After some preliminaries in \S 1, we introduce in \S 2 closed subalgebras of some type which are called function algebras induced by uniform algebras. These subalgebras contain \(C(S)\) and algebras of analytic functions. We discuss in \S 2 compactness and weak compactness of \(\varphi\) in the case \(A\) is a function algebra induced by a uniform algebra and \(\varphi\) is a composition operator. We give conditions under which \(\varphi\) is compact or weakly compact and establish the relationship between compactness and weak compactness of \(\varphi\).