The spectrum of a function algebra

Abstract

The spectrum of a function algebra A defined on a space X is sometimes obtained by attaching a thin set to X. Function algebras with this property have been constructed for various purposes by J. Wermer [5], S. J. Sidney [4], and E. Kallin [2]. We present here a theorem which reveals the underlying basis for this phenomenon. Theorem 1.1 shows that to compute the spectrum of a function algebra A on X it is sometimes sufficient to compute the spectrum of CL(A I F), where F is the set of common zeros in X of an ideal of A, and then attach it to X along F. A function algebra A is a closed point-separating subalgebra of C(X) containing the constants, where X is a compact Hausdorff space. The spectrum of A, denoted by S(A), is the compact Hausdorff space of nonzero complex homomorphisms) of A taken in the Gelfand topology. We regard X as a closed subspace of S(A). A extends via the Gelfand transform to a function algebra on S(A), and, when convenient, we shall regard A as a function algebra on S(A).