Zerofree extension of continuous functions on a compact Hausdorff space

AbstractLet CR(X) denote, as usual, the Banach algebra of all real valued continuous functions on a compact Hausdorff space X endowed with the supremum norm. We present an elementary proof of the following extension result for CR(X):For a given g∈CR(X) with zero set Zg and for the n-tuple (f1,…,fn)∈CRn(X) without common zeros in Zg the following assertions are equivalent:(i) The restriction tuple (f1,…,fn)|Zg has an extension to (F1,…,Fn)∈CRn(X) without common zeros in X.(ii) There exists an n-tuple (h1,…,hn)∈CRn(X) such that the n-tuple (f1+h1g,…,fn+hng)∈CRn(X) has no common zeros in X