Let \(C_b(K)\) be the uniform algebra of bounded continuous complex-valued functions defined on the Hausdorff space \(K.\) In Section 2, the authors study, for an arbitrary closed subspace \(A\) of \(C_b(K)\), the Gâteaux and the Fréchet differentiability of the sup-norm \(\|\cdot\|\) at a given \(f\in A.\) They show that if \(f\) is a strong peak function, then \(\|\cdot\|\) is Gâteaux differentiable at \(f\), and that the converse holds whenever \(A\) is a separating subspace, \(K\) is compact and \(f\neq 0\). As a consequence, it is deduced that for compact \(K\) and nontrivial separating separable subspaces \(A\), the set of all peak functions in \(A\) form a dense \(G_\delta\) subset of \(A\) and the set \(\rho A\) of peak points for functions in \(A\) is a norming set whose closure in \(K\) is the Shilov boundary of \(A,\) that is, the smallest closed norming subset for \(A\).
The article appears to be motivated by the study of differentiability properties of the norm of the disc algebra generalizations
\[ A_b(B_X)=\{f\in C_b(B_X): f \text{ is analytic on the interior of } B_X\}\text{ and} \]
\[ A_u(B_X)=\{f\in A_b(B_X): f \text{ is uniformly continuous}\}, \]
where \(B_X\) is the the closed ball of a nontrivial complex Banach space \(X.\) As applications of the general results obtained, it is proved that, if \(X\) has the Radon-Nikodým property, then for either \(A_b(B_X)\) or \(A_u(B_X),\) the set of strong peak functions is dense and therefore the norm is Gâteaux differentiable on a dense subset. However, the norm is nowhere Fréchet differentiable for both algebras and general \(X,\) as the authors show.
Some other similar results on the analogues for vector-valued function spaces and on the numerical Shilov boundary complete this interesting paper.