Examples of Factorization Without Bounded Approximate Units

Abstract

Examples are given of Banach algebras which do not have bounded approximate units but in which every element is a product. Another algebra is constructed in which there are elements which are not products but every element is the sum of two products. Most of the examples are commutative and separable. These examples suggest that there may be a connection between factorization questions and the topology of the carrier space of a Banach algebra.

On dense ideals of C*-algebras and generalizations of the Gelfand–Naimark Theorem

“…Our approach is particularly well suited to the study of Banach algebras having an unbounded approximate identity. That is, it is not easy to give an example of a Banach algebra with an approximate identity not bounded in the multiplier norm; Willis constructed such an algebra in [33,Example 5]. Moreover, an application of the Uniform Boundedness Principle shows that if a Banach algebra has a sequential approximate identity, then it is automatically bounded with respect to the multiplier norm; see, for instance, [14, p. 191].…”

C∗-Segal algebras with order unit

“…That is to say, it is not easy to give an example of a Banach algebra with an approximate identity not bounded in the multiplier norm. In fact, it appears that Willis was the first who constructed such an algebra in [24,Example 5]. Moreover, an application of the Uniform Boundedness Principle yields that if a Banach algebra has a sequential approximate identity, then it is automatically bounded with respect to the multiplier norm; see, for instance, [11, p. 191].…”

Unbounded approximate identities in normed algebras

“…Our counterexample to the extension of Proposition 1.2 to the unbounded case consists of some simple observations about a sophisticated example of George Willis [4]. It gives us an example which is both commutative and complete.…”