Multiplicatively spectrum-preserving maps of function algebras

Abstract

Abstract. Let X be a compact Hausdorff space and A ⊂ C(X) a function algebra. Assume that X is the maximal ideal space of A. Denoting by σ(f )

Multiplicatively Spectrum-Preserving Maps of Function Algebras. Ii

“…Let us note in passing that, in [1], an x 0 ∈ X is called a 'strong boundary point' if, given any neighbourhood U of x 0 , there exists f 1 ∈ A with f 1 ∞ = 1 and M f1 := {x : |f 1 (x)| = 1} ⊂ U (so that |f 1 | < 1 off U .) This conforms to our usage of the term 'generalized peak point' in [10]: one simply observes that if x 0 ∈ M f1 and f 1 (x 0 ) = e iθ0 , and we define g 1 = e −iθ0 f 1 ∈ A, then the function f = g 1 e g1 /e defines a peaking set P (f ) := {f = 1} ⊂ U , x 0 ∈ P (f ), and |f | < 1 off P (f ) (f ∈ A, since g 1 ∈ A, e g1 ∈ A , and A is an ideal in A ).…”

“…This is the natural extension of the theorem proved in [10], which, in turn, was a generalization of Theorem 5 in [6].…”

“…Our main concern here is with subalgebras A of C 0 (X), which are closed in the supnorm topology defined above and which are point separating in the sense that, given x, y ∈ X with x = y, there is an f ∈ A with f (x) = f (y). Such objects A we also designate as function algebras as in [10], the only difference being that the algebras considered here do not contain constants.…”

“…We define peaking functions and generalized peak points for the function algebra A as in [10]. Let us note in passing that, in [1], an x 0 ∈ X is called a 'strong boundary point' if, given any neighbourhood U of x 0 , there exists f 1 ∈ A with f 1 ∞ = 1 and M f1 := {x : |f 1 (x)| = 1} ⊂ U (so that |f 1 | < 1 off U .)…”

“…We extend and complete the results in [1] by proving in § 2 that x ∈ ∂ A (X) if and only if x is a generalized peak point (or, equivalently, a strong boundary point). The proof will be based on the following analogue, for A, of a theorem of Bishop well known in the context of uniform algebras (see (1.5) in [10] and the references given there). Theorem 1.1.…”