Closed subspaces of finite codimension in some function algebras

Abstract

ABSTRACT. We characterize all closed subspaces of finite codimension in some specific types of function algebras e.g. these include C(X): algebra of continous functions on a compact Hausdorff space, Cn [a, b]: the algebra of n-times continuously differentiable functions on the closed interval [a, b]. Our work is a generalization of the well-known Gleason-Kahane-Zelazko theorem [3,6] for subspaces of codimension one in arbitrary unitary Banach algebras.

Sufficient conditions for a linear functional to be multiplicative

“…In §2 we generalize some results in [7] and [5] and show that if X is a compact and nowhere dense subset of the plane, then R(X), the uniform closure of the algebra of rational functions with poles off X, has the P (k, n) property for all k, n ∈ N.…”

“…To prove the P (1, n) property for C(X) where X is a compact Hausdorff space, Jarosz proves a lemma [7,Lemma 1] stating that if X is a compact subset of the real line and p 1 , · · · , p n are polynomials such that each linear combination of them has a zero in X, then there is a common zero for p 1 , · · · , p n in X. Garimella and Rao [5] generalized this result for those closed subsets X of C that have an empty interior and called this result the polynomial lemma. The assumption that X has an empty interior is essential as the following example shows:…”

“…This conjecture was settled by K. Jarosz [7]. Garimella and Rao [5] showed that C n [a, b] and L 1 (R) satisfy P (k, n). Chen and Cohen [2] have proved that a selfadjoint regular n-point spectral Banach algebra satisfies P (k, n).…”