Sufficient conditions for a linear functional to be multiplicative

A commutative Banach algebra $\mathcal{A}$ is said to have the $P(k,n)$property if the following holds: Let ${{M}}$ be a closed subspace of finite codimension $n$ such that, for every $x\in {{M}}$, the Gelfand transform $\hat{x}$ has at least $k$ distinct zeros in $\Delta(\mathcal{A})$, the maximal ideal space of $\mathcal{A}$. Then there exists a subset $Z$ of $\Delta(\mathcal{A})$of cardinality $k$ such that $\hat{{M}}$ vanishes on $Z$, the set of common zeros of ${{M}}$. In this paper we show that if $X\subset \mathbf{C}$ is compact and nowhere dense, then $R(X)$, the uniform closure of the space of rational functions with poles off $X$, has the $P(k,n)$ property for all $k,n\in \mathbf{N}$. We also investigate the $P(k,n)$ property for the algebra of real continuous functions on a compact Hausdorff space.