Sufficient conditions for a linear functional to be multiplicative

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