Type: Article
Publication Date: 1979-01-01
Citations: 41
DOI: https://doi.org/10.2140/pjm.1979.80.117
A Banach algebra A is said to have "(weak) factorization" if for each feA, there exist g,heA (resp.n^l and fftihi, --,g n y h n e A) such that f=gh(f=Σgjhj).Cohen's factorization theorem says that if A has bounded approximate identity, then A has factorization.The converse is false in general.This paper investigates various implications of factorization and weak factorization for commutative algebras that are weakly self-adjoint.(Defined below; these algebras include self-adjoint algebras.)The main result is Theorem 1.3:If the weakly self-ad joint commutative Banach algebra A of functions on the locally compact space X has weak factorization, then there exists K > 0 such that, for all compact subsets E of X, there exists /ei such that 11/11 ίg K and /^ 1 on E. Applications of 1.3 are given.In particular it is shown that a proper character Segal algebra on U {G), (G a LCA group) cannot have weak factorization.We say that a Banach algebra B of complex-valued continuous functions on a topological space is weakly self-adjoint if there exists iΓ 0 > 0 such that for each / e B (0.1) ]/| 2 ei? and \\\f\*\\ B ^K0 \\f\\%.adjoint Banach algebra A of complex-valued continuous functions.Recall that B is called a Banach ideal of a Banach algebra A if \\f\\ B ^ \\f\\ A and fg e B, with \\fg\\ B Ĥ/IUIfML f°r a ^ f eB, g eA.Dense Banach ideals have been called A-Segal algebras by Burnham [1] and others.It is worth mentioning that a self-adjoint Banach algebra B has weak factorization if and only if each element of B can be written as a linear combination of elements of the form | /| 2 , / e B (this follows easily from the identity Our work was motivated by an attempt to give a converse to Cohen's theorem for Segal algebras on abelian groups (defined in §2), and thereby extend the results of Burnham [1], Leinert [7], Wang [15], Yap [17] and others.Since such a Segal algebra cannot have bounded approximate units, that would prove that a proper Segal algebra cannot have weak factorization.We cannot prove that in full generality, but we are able to improve earlier results substantially.