The Bézout properties for some classical function algebras (Q1928398)

In this contribution to the longstanding study of divisibility in Banach algebras, the authors address two essential questions in relation to certain classical algebras of functions defined on a compact set in the complex plane: (i) Given two elements of the algebra, do they have a greatest common divisor in the algebra? (ii) If, for some elements \(a\) and \(b\) in the algebra, a greatest common divisor exists, then is \(\text{gcd}(a,b)\) in the ideal generated by \(a\) and \(b\)? A commutative unital ring \(R\) is called a pseudo-Bézout ring provided that any two non-zero elements have a gcd in \(R\) and is a pre-Bézout ring provided that, whenever it exists, \(\text{gcd}(a,b)\) is in the ideal generated by \(a\) and \(b\). A ring that has both the pseudo- and pre-Bézout properties is called simply a Bézout ring. (Some technicalities that arise when the ring has zero-divisors are carefully addressed in the paper, but will be glossed over here.) For a compact set \(K\) in the complex plane, let \(C(K)\) denote the algebra of complex-valued functions that are continuous on \(K\), let \(P(K)\) denote the uniform closure of the polynomials in \(C(K)\), \(R(K)\) the smallest closed subalgebra of \(C(K)\) containing the rational functions with poles off of \(K\), and \(A(K)\) the algebra of continuous functions on \(K\) that are holomorphic in the interior of \(K\). Theorem 3.1 asserts that, if \(K\) is an infinite set and \(A\) is any uniformly closed subalgebra of \(C(K)\) that contains \(P(K)\), then \(A\) is not pseudo-Bézout. For the proof, the authors show the existence of a function \(g\in P(K)\) of the form \(g=(z-\alpha)h\), where \(\alpha\) is a non-isolated peak point for \(P(K)\) and \(h\) has a discontinuity at \(\alpha\). Then it is shown that the functions \((z-\alpha)\) and \(g\) have no greatest common divisor in \(A\). In Theorem 3.7, the authors show that, if \(K\) is compact and both \(K\) and its interior \(K^\circ\) are connected and there exists a positive number \(\kappa>0\) such that every component \({\mathbb C}\backslash K\) has diameter larger than \(\kappa\), then \(A(K)\) and \(R(K)\) are both pre-Bézout rings. Section 4 of the paper examines the function algebra \(H^\infty +C\) and finds, in Theorem 4.1, that there are pairs of functions in this algebra that have no greatest common divisor. Hence, \(H^\infty +C\) is not pseudo-Bézout. The focus in Section 6 is on the algebras \(C(X,\tau)\), where \(X\) is a compact Hausdorff space with topological involution \(\tau\). These algebras are always pre-Bézout (Theorem 6.1), but may or may not be pseudo-Bézout depending on the nature of the set \(X\) (see Propositions 6.6 and 6.12). Other sections of the paper look at algebras between the disk algebra and \(H^\infty({\mathbb D})\); spaces of functions of several (complex) variables; and connections between the Bézout properties and the Bass stable rank of a Banach algebra.