Let A be the disk algebra, of continuous functions on the closed disk holomorphic on the interior, with sup-norm denoted || . || . Let x be an interior point of the disk. Does there exist a proper A-ideal J such that the sup over nonzero elements of J of | f(x) | / || f || is 1 ?



ideals in the disk algebra
How about the following: let $M\_1=\{f\in A : f(1)=0\}$, which is a maximal ideal in $A$, and let $f\_n(z) = (1-z)^{1/n} = \exp (n^{-1}\log(1-z))$ where we take $(-\infty,0]$ as our branch cut for $\log$ (the point being that $z\mapsto 1-z$ maps the open unit disc into the open right-half-plane).


Then $(f\_n)$ is a sequence in $M\_1$, $\Vert f\_n\Vert\_\infty = 2^{1/n}$, and $f\_n(0)=1$ for all $n$, so
$$ 1\geq \sup \{ |f(0)| : f\in M\_1, \Vert f\Vert\_\infty=1\} \geq \sup\_n 2^{-1/n} |f\_n(0)| = 1$$
which seems to be what you wanted.


(Of course there was nothing special about $0$ as a choice of interior point, since we can always post-compose with an appropriate Möbius transformation to move any prescribed interior point to $0$ while not affecting norms in the disk algebra.)


