Vol. 80, No. 1, 1979

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Nonfactorization in commutative, weakly selfadjoint Banach algebras

Hans Georg Feichtinger, Colin C. Graham and Eric Howard Lakien

Vol. 80 (1979), No. 1, 117–125

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 fK 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.

Mathematical Subject Classification 2000
Primary: 46J10
Secondary: 43A15
Received: 28 April 1977
Revised: 28 December 1977
Published: 1 January 1979
Hans Georg Feichtinger
Colin C. Graham
1115 Lenora Road
Bowen Island BC V0N 1G0
Eric Howard Lakien