On an Algebraic Extension of $A(E)$

An algebraic extension of the algebra $A(E)$, where $E$ is a compactum in $\mathbb C$ with nonempty connected interior, leads to a Banach algebra $B$ of functions that are holomorphic on some analytic set $K^\circ \subset \mathbb C^2$ with boundary $bK$ and continuous up to $bK$. The singular points of the spectrum of $B$ and their defects are investigated. For the case in which $B$ is a uniform algebra, the depth of $B$ in the algebra $C(bK)$ is estimated. In particular, conditions under which $B$ is maximal on $bK$ are obtained.