Generalized stable ranks for commutative Banach algebras

Abstract

We introduce the notion of generalized E-stable ranks for commutative unital Banach algebras and determine these ranks for the disk-algebra \({A(\mathbb{D})}\), many of its subalgebras, and the algebra H ∞ of bounded holomorphic functions in the unit disk. Relations to L-sets and separating algebras, notions due to Csordas and Reiter, are given, too. Finally we show that the absolute stable rank of \({A(\mathbb{D})}\) and H ∞ is bigger than 2.