Factorization in commutative Banach algebras

Abstract

We consider when a (non-unital) commutative Banach algebra has a variety of factorization properties; we list the (obvious) implications between these properties, and then investigate whether any of these implications can be reversed in various classes of commutative Banach algebras. We summarize the known counter-examples to these possible reverse implications, and add further counter-examples. Some results are used to show the existence of a large family of prime ideals in each non-zero, commutative, radical Banach algebra with a dense set of products.

Extensions of uniform algebras

“…A more recent construction of a generalised Cole extension is given in [11], where uniform algebras in which certain null sequences factor. We summarise some of the details of their construction below.…”

“…In fact, B is a generalised Cole extension of A implemented by T. Moreover, the fibre of Π over x 0 consists of a single point. (Extensions such as the above where there is a point x 0 whose fibre consists of a single point are called distinguished point extensions in [11]. )…”

“…Many recent examples in uniform algebra theory can be realised-or are explicitly constructed-as uniform algebra extensions. For example, in [11], Dales, Feinstein, and Pham construct a uniform algebra that contains a maximal idea in which all null sequences factor but does not contain a bounded approximate identity. In [18], Feinstein and Izzo construct give a construction that yields essential uniform algebras.…”