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Bishop’s theorem and differentiability of a subspace of ${C}_{b}\left(K\right)$. (English) Zbl 1217.46015

Let ${C}_{b}\left(K\right)$ 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 ${C}_{b}\left(K\right)$, the Gâteaux and the Fréchet differentiability of the sup-norm $\parallel ·\parallel$ at a given $f\in A·$ They show that if $f$ is a strong peak function, then $\parallel ·\parallel$ is Gâteaux differentiable at $f$, and that the converse holds whenever $A$ is a separating subspace, $K$ is compact and $f\ne 0$. 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

${A}_{b}\left({B}_{X}\right)=\left\{f\in {C}_{b}\left({B}_{X}\right):f\phantom{\rule{4.pt}{0ex}}\text{is}\phantom{\rule{4.pt}{0ex}}\text{analytic}\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{interior}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}{B}_{X}\right\}\phantom{\rule{4.pt}{0ex}}\text{and}$
${A}_{u}\left({B}_{X}\right)=\left\{f\in {A}_{b}\left({B}_{X}\right):f\phantom{\rule{4.pt}{0ex}}\text{is}\phantom{\rule{4.pt}{0ex}}\text{uniformly}\phantom{\rule{4.pt}{0ex}}\text{continuous}\right\},$

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 ${A}_{b}\left({B}_{X}\right)$ or ${A}_{u}\left({B}_{X}\right),$ 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.

##### MSC:
 46E15 Banach spaces of continuous, differentiable or analytic functions 46G05 Derivatives, etc. (functional analysis) 49J50 Fréchet and Gateaux differentiability 46G20 Infinite dimensional holomorphy
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