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 whenever is approximately regular.

     

On the invertible algebras linear over an Abelian group

In this paper using the second order formula, namely the ??(?)-identities, we characterize some subclasses of the invertible algebras that are linear over an Abelian group and have restrictions on the use of the automorphisms of the corresponding group.     

Algebraic properties of isometries between groups of invertible elements in Banach algebras

We prove that an isometry T between open subgroups of the invertible groups of unital Banach algebras A and B is extended to a real-linear isometry up to translation between these Banach algebras. While a unital isometry between unital semisimple commutative Banach algebras need not be multiplicative, we prove in this paper that if A is commutative and A or B are semisimple, then (T(eA))⁻¹T is extended to an isometric real algebra isomorphism from A onto B. In particular, A−1 is isometric as a metric space to B−1 if and only if they are isometrically isomorphic to each other as metrizable groups if and only if A is isometrically isomorphic to B as a real Banach algebra; it is compared by the example of ?elazko concerning on non-isomorphic Banach algebras with the homeomorphically isomorphic invertible groups. Isometries between open subgroups of the invertible groups of unital closed standard operator algebras on Banach spaces are investigated and their general forms are given.     

On amalgamated Banach algebras

If F(z) is a polynomial of degree n having all zeros in \(|z|\le k,~k>0\) and f(z) is a polynomial of degree \(m\le n\) such that \(|f(z)|\le |F(z)|\) for \(|z|=k\), then it was formulated by Rather and Gulzar (Adv Inequal Appl 2:16–30, 2013) that for every \(|\delta |\le 1, |\beta |\le 1,~R>r\ge k\) and \(|z|\ge 1,\)

$$\begin{aligned} |B[fo\sigma ](z)+\psi B[fo\rho ](z)|\le |B[Fo\sigma ](z)+\psi B[Fo\rho ](z)|, \end{aligned}$$

where B is a \(B_{n}\) operator, \(\sigma (z){=}Rz, \rho (z){=}rz\) and \(\psi {:=}\psi (R,r,\delta ,\beta ,k) {=}\beta \bigg \{\bigg (\frac{R+k}{r+k}\bigg )^{n}{-}|\delta |\bigg \}{-}\delta \). The authors have assumed that \(B\in B_{n}\) is a linear operator which is not true in general. In this paper, besides discussing assumption of authors and their followers (see e.g, Rather et al. in Int J Math Arch 3(4):1533–1544, 2012), we present the correct proof of the above inequality. Moreover our result improves many prior results involving \(B_{n}\) operators and a number of polynomial inequalities can also be deduced by a uniform procedure.

     

On automorphisms of Banach algebras

We prove an extension of the Jacobson density theorem for Banach algebras with automorphisms, and discuss its applications. In particular, automorphisms a \alpha of Banach algebras such that the map a- 1 \alpha - 1 is spectrally bounded are characterized.     

On identifying the maximal ideals in Banach algebras

We consider the problem of identifying the maximal ideals of a Banach algebra S that is contained in a larger Banach algebra B. If S is dense in B in an appropriate sense and if the spectral radii in S and B are the same, then S and B have the same maximal ideals. The result is illustrated by two examples of Banach algebras S in which the density and spectral radius conditions are easily shown to be valid with respect to a larger algebra B whose maximal ideals are known. The first example is a convolution algebra on a group where the functions in the algebra have specified rate of decay. The second example is a generalized version of an algebra introduced by I. Hirschman. Two applications of the generalized Hirschman algebra are presented: these relate to filtering and prediction of stationary random sequences and concern the asymptotic behavior of the errors incurred by the finite memory predictor and by the finite lag filter.     

On the symbolic calculus in homogeneous Banach algebras

We construct a strongly homogeneous Banach algebraB so that for an appropriate positive integern, (1)A(T) ⊊B ⊊C(T); (2)n-times continuously differentiable functions operate onB. This work was supported in part by NSF Grant MCS75-08552-A02.     

A characterizing property of commutative Banach algebras may not be sufficient only on the invertible elements

In this article, we construct a commutative unital Banach algebra, in which the property $6a^{2}6={6a6}^2$ is true for the invertible elements, but cannot be extended to the whole algebra.     

On character amenability of Banach algebras

We continue our work [E. Kaniuth, A.T. Lau, J. Pym, On φ-amenability of Banach algebras, Math. Proc. Cambridge Philos. Soc. 144 (2008) 85-96] in the study of amenability of a Banach algebra A defined with respect to a character φ of A. Various necessary and sufficient conditions of a global and a pointwise nature are found for a Banach algebra to possess a φ-mean of norm 1. We also completely determine the size of the set of φ-means for a separable weakly sequentially complete Banach algebra A with no φ-mean in A itself. A number of illustrative examples are discussed.     

Regular Fréchet-Lie groups of invertible elements in some inverse limits of unital involutive Banach algebras

We consider a wide class of unital involutive topological algebras provided with aC ^*-norm and which are inverse limits of sequences of unital involutive Banach algebras; these algebra sare taking a prominent position in noncommutative differential geometry, where they are often called unital smooth algebras. In this paper we prove that the group of invertible elements of such a unital solution smooth algebra and the subgroup of its unitary elements are regular analytic Fréchet-Lie groups of Campbell-Baker-Hausdorff type and fulfill a nice infinite-dimensional version of Lie's second fundamental theorem.     

On the semi-continuity of generalized inverses in Banach algebras

The main purpose of this paper is to study the continuity of several kinds of generalized inverses of elements in a Banach algebra with identity. We first obtain a sufficient and necessary condition for the lower semi-continuity of reflexive generalized inverses as set-valued mappings. Based on this result, we characterize the continuity of the Moore-Penrose inverse in a C^∗-algebra and therefore, derive some new and well-known criteria in operator theory.     

On the simplicial cohomology of Banach operator algebras

We investigate the simplicial cohomology of certain Banach operator algebras. The two main examples considered are the Banach algebra of all bounded operators on a Banach space and its ideal of approximable operators. Sufficient conditions will be given forcing Banach algebras of this kind to be simplicially trivial.     

On commutativity and the numerical range in Banach algebras

If x and y are noncommuting elements in a complex Banach algebra with unit then the generalized numerical ranges W₀(e^λvxe^−λv) cover the complex plane.