Projective free algebras of continuous functions on compact abelian groups

Abstract

It is proved that the Wiener algebra of functions on a connected compact abelian group whose BohrFourier spectra are contained in a fixed subsemigroup of the (additive) dual group, is projective free. The semigroup is assumed to contain zero and have the property that it does not contain both a nonzero element and its opposite. The projective free property is proved also for the algebra of continuous functions with the same condition on their Bohr-Fourier spectra. As an application, the connected components of the set of factorable matrices are described. The proofs are based on a key result on homotopies of continuous maps on the maximal ideal spaces of the algebras under consideration.

On Stable Rank of ${\mathbf H^\infty}$ on Coverings of Finite Bordered Riemann Surfaces

“…In particular, each matrix in SL n (H ∞ (S ′ )) can be presented as a finite product of elementary matrices. (This is a well-known fact; it can be deduced, e.g., from [BRS,Thm. 2.1].)…”

On the Factorization of Matrices Over Commutative Banach Algebras

“…(It easily follows from [T,Thm. 1] that the Bass stable rank of W (R) Γ is 1 and from [BRS,Thm. 1.2] that H 1 (M(W (R) Γ ), Z) = 0.)…”

On Homotopy Invariants of Tensor Products of Banach Algebras

“…1.2]. In particular, this is valid for continuous maps ofM(A) into Y ∈ B −1 , B −1 l , id B .Using this and repeating literally the arguments of the proofs of[BRS, Thm. 5.1, Thm.…”

“…Proof of Theorem 1.4. Since A ∈ C , every continuous map of M(A) into an absolute neighbourhood retract Y is homotopic to a constant map into Y , see the argument of the proof of[BRS, Thm. 1.2].…”