Factorization and unbounded approximate identities in Banach algebras

by P.G.Dixon

Math. Proc. Camb. Philos. Soc., 107 (1990), 557-571.

1985 Mathematics Subject Classification: primary 46J05; secondary 46J10, 46J20.


Cohen's Factorization Theorem says, in its basic form, that if A is a Banach algebra with a bounded left approximate identity, then every element x \in A may be written as a product x = ay for some a,y \in A. Such is the beauty and importance of this result that much interest attaches to the question of whether the hypothesis of a bounded left approximate identity can be weakened, or whether a converse result exists. This paper contributes to the study of that question.

A stronger form of Cohen's Theorem than that stated above has as its conclusion that there exists K>1 such that for every \epsilon > 0 every x \in A can be written in the form x=ay with ||a|| < K and ||y - x|| < \epsilon. It is easily shown that this stronger factorization statement implies the existence of a bounded left approximate identity, (Doran and Wichmann's book, Theorem (16.3)).

Another strong form of Cohen's Theorem has the conclusion that, for every sequence x_1,x_2,... in A with x_n \to 0, there exist a,y_1,y_2,... \in A with y_n \to 0 and x_n = ay_n for all n. We call this ``factorization of null sequences''.

Does factorization of null sequences imply the existence of a bounded left approximate identity? If A is non-commutative there is no hope of this: the four-dimensional algebra A = (C^4,||.||_1) with multiplication

(x_1,x_2,x_3,x_4)(y_1,y_2,y_3,y_4) = (x_1y_1,x_1y_2,x_3y_3,x_4y_3)

has factorization of null sequences, but neither left nor right approximate units, since, if x = (0,1,0,1) then ||xu - x|| > 1 and ||ux - x|| > 1, for all u \in A. (This is unsurprising: the algebra is constructed by taking the direct sum of an algebra with just a left identity and an algebra with just a right identity.)

For commutative A, we have the example of Leinert (see Doran and Wichmann's book, Example(22.6)) of a commutative semisimple Banach algebra (a semigroup algebra, in fact) with factorization of null sequences but without approximate units. We also have an example of Ouzomgi consisting of a maximal ideal of H^\infty of the disc corresponding to a character of H^\infty which is a one-point Gleason part and does not belong to the Shilov boundary; this is a commutative, semisimple Banach algebra (in this case, a uniform algebra), with factorization of single elements, but without a bounded approximate identity. However, neither of these examples is separable.

Recently, G. A. Willis obtained an example of a commutative, separable Banach algebra, with factorization of null sequences but no bounded approximate identity. This example solved a major outstanding problem, but it has a significant defect. Unlike the two non-separable examples just mentioned, its multiplier norm is not equivalent to its original norm.* It therefore poses the problem of whether factorization of null sequences, or even just factorization of single elements, implies the existence of a bounded approximate identity in the presence of such extra conditions.

In the present paper, we show,(Theorem 3.1), that every commutative separable Banach algebra with factorization of null sequences has a (possibly unbounded) approximate identity. Further, this approximate identity may be chosen so that its Gelfand transforms are bounded by arbitrarily slowly growing functions on the maximal ideal space.

This raises the question of when we can deduce the existence of a bounded approximate identity from the existence of an approximate identity with very weak boundedness properties. We shall show (Theorem4.1) that if A is a commutative Banach algebra and, for every x \in A, there exist constants K > 0, \alpha \in (0,1/2) such that, for each \epsilon \in (0,1), there is a u \in A with ||u|| < K\epsilon^{-\alpha} and ||ux - x|| < \epsilon||x||, then A has a bounded approximate identity. This ``conditional boundedness'' property is inspired by work of Feichtinger and Leinert on factorization of individual elements in Banach algebras.

Section 5 applies these ideas to maximal ideals in uniform algebras and presents a new proof that if a maximal ideal has a bounded approximate identity, then it corresponds to a strong boundary point.

In section 6 we specialize to maximal ideals in the algebras R(X) for compact plane sets X, where we have Melnikov's characterization of strong boundary points at our disposal. We show that, in this case, factorization of null sequences does imply the existence of a bounded approximate identity. The paper concludes with some open questions.

* After this paper was written, Willis obtained an example without this defect


Reprints of the paper available on request: e-mail P.Dixon with extension  @sheffield.ac.uk.

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