Dense morphisms in commutative Banach algebras

Abstract

Abstract. Using a new notion of stability we compute exactly the stable rank of the polydisc algebra, extend Oka's extension theorem to «-tuples of functions without common zeros and give an estimation for a question raised by Swan concerning the stable rank of a dense subalgebra of a given Banach algebra.

The Stable Rank of Topological Algebras and a Problem of R. G. Swan

“…Also [CoSu87], a (dense) onto morphism f : A Ä B of commutative unital Banach algebras induces an (approximate) Serre fibration f n : Lg n (A) Ä Lg n (B). We will need corresponding facts for the more general situation of module morphisms.…”

“…The idea of the above proof is taken from [CoSu87]. It can be proved that T is a principal fiber bundle (cf.…”

Approximation by invertible elements and the generalized $E$-stable rank for $A({\boldsymbol D})_{\mathsf R}$ and $C({\boldsymbol D})_{\mathrm{sym}}$

“…Let us recall that if f ∈ A(D) does not vanish on the polynomial convex set E, then f −e p n E → 0 for some sequence of polynomials (see [7] and [13]). Now a first attempt at a proof in A(D) R could work as follows: let p n be polynomials with e p n − f E → 0.…”

“…Recall that the dense stable rank (or approximate stable rank) of A(D) is one ( [7], [13]) and that a compact set K ⊆ D is polynomial convex (or equivalently A(D)-convex) if and only if K has no holes.…”

Approximation by invertible functions of $H^{\infty}$

“…, a n + b n a n+1 ) ∈ U n (A)}, with Bsr A = ∞ if there is no such n. An alternative and equivalent definition is the minimum integer n such that for every onto morphism of Banach (or uniform) algebras f : A → B, the induced map from U n (A) into U n (B) is onto (see [4]). This led naturally to define another invariant of uniform algebras, where onto morphisms are replaced by morphisms with dense image [5]. Specifically, the dense stable rank of A (dsr A) is the minimum n such that for every morphism of uniform algebras f : A → B with dense image, the induced application from U n (A) into U n (B) has dense image.…”