Characters on Algebras of Vector-Valued Continuous Functions

Abstract

Let A be a topological algebra with continuous multiplication, X a completely regular Hausdorff space and C(X, A) the algebra of all A-valued continuous functions on X. We describe the characters on subalgebras of C(X, A) by means of those of A and evaluations at points of the Stone-Čhech compactification of X.

Some generalizations of the Hewitt extension

“…Some cases when the space of possible values of p might be some other subspace of βX have been considered by L. Oubbi. In what follows we shall state some of his results using the extension α A X and fix a small error in a lemma in his article [12].…”

“…In [12], subalgebras A ⊂ C(X, Y ), where Y is an algebra with continuous multiplication over F = C, are considered and the following subsets of βX are defined:…”

“…It is claimed in the first part of Lemma 2.1 in [12] that if A is a Y -bimodule and a C b (X, R)-module (i. e. closed under pointwise multiplication by elements of Y and C b (X, R)), then the maps…”

“…So, if the algebra Y has at least two continuous characters then the values of G 1 and G 2 are not always characters, and if, in addition, the space X has at least two points then they are not one-to-one either. The statement can be fixed and the proof in [12] works, if the maps G 1 and G 2 are regarded with different domains (the proof also works if F = R or if multiplication on Y is not jointly continuous): Proposition 6.1. Let X be a Tychonoff space, Y a Haudorff topological algebra over F ∈ {R, C} and A ⊂ C(X, Y ) a function algebra that is a Y -bimodule and a C b (X, R)-module.…”

“…The second part of Lemma 2.1 in [12] says that G 1 is continuous if Hom Y is locally equicontinuous; this works for algebras over any topological field, too.…”