Homomorphisms on a class of commutative Banach algebras (Q350767)

Let \(X\) be a compact connected metric space. Let \({\mathcal B}\) be either the space of convergent (\(c\)) or bounded sequences (\(\ell^{\infty}\)). Let NEWLINE\[NEWLINE\mathrm{Lip}_*(X,B) = \Bigl\{ f: X \rightarrow {\mathcal B}: \sup_{x \neq y} \frac{\|f(x)-f(y)\|}{d(x,y)} < \infty\Bigr\}.NEWLINE\]NEWLINE Equipped with the norm \(\|f\|_* = \|f\|+ \sup_{x \neq y} \frac{\|f(x)-f(y)\|}{d(x,y)} \) and the canonical \(*\)-operation, this space is a commutative Banach \(*\)-algebra. In this interesting paper, the authors study the structure of homomorphisms \(\psi\) (for domains \(X,Y\)) that fix the constant functions, showing in particular that they are automatically continuous.NEWLINENEWLINEIn the case of \(c\), there exists a continuous map \(\tau\) of the one-point compactification of the positive integers and a sequence of Lipschitz maps \(\phi_{n,\tau(n)}: Y \rightarrow X\), such that \(\psi(f)_n(y) = f_{\tau(n)} \circ \phi_{n,\tau(n)}(y)\) for all \(y,n\).NEWLINENEWLINEIn the case of \(\ell^{\infty}\), the map \(\tau\) is defined on the Stone-Čech compactification.