A Banach algebra A is said to
have “(weak) factorization” if for each f ∈ A, there exist g,h ∈ A (resp. n ≧ 1 and
g1,h1,⋯,gn,hn∈ A) such that f = gh (f =∑gjhj). 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-adjoint 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 f ∈ A such that
∥f∥≦ K and f ≧ 1 on E. Applications of 1.3 are given. In particular it is shown that
a proper character Segal algebra on L1(G), ( G a LCA group) cannot have weak
factorization.
