## Factorization in commutative Banach algebras

### Volume 259 / 2021

Studia Mathematica 259 (2021), 79-120
MSC: Primary 46J10; Secondary 46J15, 46J20, 46J45.
DOI: 10.4064/sm191216-22-7
Published online: 1 February 2021

#### 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.