J. F. Feinstein

On the denseness of the invertible group in Banach algebras

We examine the condition that a complex Banach algebra A has dense invertible group. We show that, for commutative algebras, this property is preserved by integral extensions. We also investigate the connections with an old problem in the theory of uniform algebras.

Trivial Jensen measures without regularity

In this note we construct Swiss cheeses X such that R(X) is non-regular but such that R(X) has no non-trivial Jensen measures. We also construct a non-regular uniform algebra with compact, metrizable character space such that every point of the character space is a peak point.

Banach function algebras with dense invertible group

In 2003 Dawson and Feinstein asked whether or not a Banach function algebra with dense invertible group can have a proper Shilov boundary. We give an example of a uniform algebra showing that this can happen, and investigate the properties of such algebras. We make some remarks on the topological stable rank of commutative, unital Banach algebras. In particular, we prove that

Compact Endomorphisms of Banach Algebras of Infinitely Differentiable Functions

\mathrm{tsr}(A) \geq \mathrm{tsr}(C(\Phi_A))

Strong Ditkin algebras without bounded relative units

whenever

A NOTE ON IDEAL SPACES OF BANACH ALGEBRAS

A

Quasicompact endomorphisms of commutative semiprime Banach algebras

is approximately regular.

A note on strong Ditkin algebras

In this note we study the endomorphisms of certain Banach algebras of infinitely differentiable functions on compact plane sets, associated with weight sequences M. These algebras were originally studied by Dales, Davie and McClure. In a previous paper this problem was solved in the case of the unit interval for many weights M. Here we investigate the extent to which the methods used previously apply to general compact plane sets, and introduce some new methods. In particular, we obtain many results for the case of the closed unit disc. This research was supported by EPSRC grant GR/M31132

A counterexample to a conjecture of S. E. Morris

In a previous note the author gave an example of a strong Ditkin algebra which does not have bounded relative units in the sense of Dales. In this note we investigate a certain family of Banach function algebras on the one point compactification of ℕ, and see that within this family are many easier examples of strong Ditkin algebras without bounded relative units in the sense of Dales.

A Non-Trivial, Strongly Regular Uniform Algebra

In a previous paper the second author introduced a compact topology on the space of closed ideals of a unital Banach algebra A. If A is separable then this topology is either metrizable or else neither Hausdorff nor first countable. Here it is shown that this topology is Hausdorff if A is the algebra of once continuously differentiable functions on an interval, but that if A is a uniform algebra then this topology is Hausdorff if and only if A has spectral synthesis. An example is given of a strongly regular, uniform algebra for which every maximal ideal has a bounded approximate identity, but which does not have spectral synthesis.