2012/08/02 by Mortaza Abtahi, Abtahi, Mortaza
Mathematics · #46J10 #46J20 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:46J10 #msc:46J20
paper · pdf · doi:10.48550/arxiv.1208.0406
arxiv created 2012/08/02 · arxiv updated 2012/08/03
Let X be a compact Hausdorf space, let A be a commutative unital Banach algebra, and let C(X,A) denote the algebra of continuous A-valued functions on X equipped with the uniform norm ||f||=sup||f(x)||:x∈ X for all f in C(X,A). Hausner, in [Proc. Amer. Math. Soc. 8(1957), 246--249], proved that M is a maximal ideal in C(X,A) if and only if there exist a point x in X and a maximal ideal N in A such that M=f in C(X,A) : f(x) in N. In this note, we give new characterizations of maximal ideals in C(X,A). We also present a short proof of Hausner's result by a different approach.