Let $H^\infty $ be the algebra of bounded holomorphic functions on the open unit disk, and let $\mathfrak M$ be its maximal ideal space. Let $\mathfrak M_a$ be the union of nontrivial Gleason parts (analytic disks) of $\mathfrak M$. In this paper, we study the problem of extensions of bounded Banach-valued holomorphic functions and holomorphic maps with values in Oka manifolds from Gleason parts of $\mathfrak M_a\setminus \mathbb {D}$. The resulting extensions satisfy the uniform boundedness principle in the sense that their norms are bounded by constants that do not depend on the choice of the Gleason part. The results extend fundamental results of D. Suárez on the characterization of the algebra of restrictions of Gelfand transforms of functions in $H^\infty $ to Gleason parts of $ \mathfrak M_a\setminus \mathbb {D}$. The proofs utilize our recent advances on $\bar \partial $-equations on quasi-interpolating sets and Runge-type approximations.

Let A be a Banach algebra and let X be a Banach A-bimodule. We consider the Banach algebra ${A\oplus _1 X}$ , where A is a commutative Banach algebra. We investigate the Bochner–Schoenberg–Eberlein (BSE) property and the BSE module property on $A\oplus _1 X$ . We show that the module extension Banach algebra $A\oplus _1 X$ is a BSE Banach algebra if and only if A is a BSE Banach algebra and $X=\{0\}$ . Furthermore, we consider $A\oplus _1 X$ as a Banach $A\oplus _1 X$ -module and characterise the BSE module property on $A\oplus _1 X$ . We show that $A\oplus _1 X$ is a BSE Banach $A\oplus _1 X$ -module if and only if A and X are BSE Banach A-modules.

We consider the algebra of holomorphic functions on L∞ that are symmetric, i.e. that are invariant under composition of the variable with any measure-preserving bijection of [0, 1]. Its spectrum is identified with the collection of scalar sequences such that is bounded and turns to be separable. All this follows from our main result that the subalgebra of symmetric polynomials on L∞ has a natural algebraic basis.

For a locally compact group $G$ with left Haar measure and a Young function ${\rm\Phi}$, we define and study the weighted Orlicz algebra $L_{w}^{{\rm\Phi}}(G)$ with respect to convolution. We show that $L_{w}^{{\rm\Phi}}(G)$ admits no bounded approximate identity under certain conditions. We prove that a closed linear subspace $I$ of the algebra $L_{w}^{{\rm\Phi}}(G)$ is an ideal in $L_{w}^{{\rm\Phi}}(G)$ if and only if $I$ is left translation invariant. For an abelian $G$, we describe the spectrum (maximal ideal space) of the weighted Orlicz algebra and show that weighted Orlicz algebras are semisimple.

We recently introduced a weighted Banach algebra $\mathcal{A}_{G}^{n}$ of functions that are holomorphic on the unit disc $\mathbb{D}$, continuous up to the boundary, and of the class ${{C}^{\left( n \right)}}$ at all points where the function $G$ does not vanish. Here, $G$ refers to a function of the disc algebra without zeros on $\mathbb{D}$. Then we proved that all closed ideals in $\mathcal{A}_{G}^{n}$ with at most countable hull are standard. In this paper, on the assumption that $G$ is an outer function in ${{C}^{\left( n \right)}}\,\left( {\bar{\mathbb{D}}} \right)$ having infinite roots in $\mathcal{A}_{G}^{n}$ and countable zero set ${{h}_{o}}\left( G \right)$, we show that all the closed ideals $I$ with hull containing ${{h}_{o}}\left( G \right)$ are standard.

Let COP = 0 ∩ H∞, where 0 is the little Bloch space on the open unit disk , and A() be the disk algebra on . For non-zero functions u1,u2,. . ., uN ∈ A() and distinct analytic self-maps ϕ1,ϕ2,. . .,ϕN satisfying ϕj ∈ A() and ∥ϕj∥∞=1 for every j, it is given characterisations of which the sum of weighted composition operators ∑Nj=1ujCϕj maps COP into A().

In the spectrum of the algebra of symmetric analytic functions of bounded type on ℓp, 1 ≤ p < +∞, and along the same lines as the general non-symmetric case, we define and study a convolution operation and give a formula for the ‘radius’ function. It is also proved that the algebra of analytic functions of bounded type on ℓ1 is isometrically isomorphic to an algebra of symmetric analytic functions on a polydisc of ℓ1. We also consider the existence of algebraic projections between algebras of symmetric polynomials and the corresponding subspace of subsymmetric polynomials.

The Banach convolution algebras ${{l}^{1}}(\omega )$ and their continuous counterparts ${{L}^{1}}\left( {{\mathbb{R}}^{+}},\omega \right)$ are much studied, because (when the submultiplicative weight function $\omega $ is radical) they are pretty much the prototypic examples of commutative radical Banach algebras. In cases of “nice” weights $\omega $, the only closed ideals they have are the obvious, or “standard”, ideals. But in the general case, a brilliant but very difficult paper of Marc Thomas shows that nonstandard ideals exist in ${{l}^{1}}(\omega )$. His proof was successfully exported to the continuous case ${{L}^{1}}\left( {{\mathbb{R}}^{+}},\omega \right)$ by Dales and McClure, but remained difficult. In this paper we first present a small improvement: a new and easier proof of the existence of nonstandard ideals in ${{l}^{1}}(\omega )$ and ${{L}^{1}}\left( {{\mathbb{R}}^{+}},\omega \right)$. The new proof is based on the idea of a “nonstandard dual pair” which we introduce. We are then able to make a much larger improvement: we find nonstandard ideals in ${{L}^{1}}\left( {{\mathbb{R}}^{+}},\omega \right)$ containing functions whose supports extend all the way down to zero in ${{\mathbb{R}}^{+}}$, thereby solving what has become a notorious problem in the area.

On se donne un intervalle ouvert non vide $\omega $ de $\mathbb{R}$, un ouvert connexe non vide $\Omega $ de ${{\mathbb{R}}_{5}}$ et unpolynôme unitaire

$${{P}_{m}}\left( z,\text{ }\!\!\lambda\!\!\text{ } \right)={{z}^{m}}+{{a}_{1}}\left( \text{ }\!\!\lambda\!\!\text{ } \right){{z}^{m-1}}=+\cdot \cdot \cdot +{{a}_{m-1}}\left( \text{ }\!\!\lambda\!\!\text{ } \right)z+{{a}_{m}}\left( \text{ }\!\!\lambda\!\!\text{ } \right)$$

de degré $m>0$, dépendant du paramètre $\lambda \in \Omega $. Un tel polynôme est dit $\omega $-hyperbolique si, pour tout $\lambda \in \Omega $, ses racines sont réelles et appartiennent à $\omega $.

On suppose que les fonctions ${{a}_{k}},k=1,\cdot \cdot \cdot ,m$, appartiennent à une classe ultradifférentiable ${{C}_{M}}\left( \Omega \right)$. On s‘intéresse au problème suivant. Soit $f$ appartient à ${{C}_{M}}\left( \Omega \right)$, existe-t-il des fonctions ${{Q}_{f}}$ et ${{R}_{f,k}},k=0,\cdot \cdot \cdot ,m-1$, appartenant respectivement à ${{C}_{M}}\left( \omega \,\times \,\Omega \right)$ et à ${{C}_{M}}\left( \Omega \right)$, telles que l’on ait, pour $\left( x,\,\lambda \right)\,\in \,\omega \,\times \,\Omega$,

$$f\left( x \right)={{P}_{m}}\left( x,\text{ }\!\!\lambda\!\!\text{ } \right){{Q}_{f}}\left( x,\text{ }\!\!\lambda\!\!\text{ } \right)+\sum\limits_{k=0}^{m-1}{{{x}^{k}}{{R}_{f,k}}\left( \text{ }\!\!\lambda\!\!\text{ } \right)?}$$

On donne ici une réponse positive dès que le polynôme est $\omega$-hyperbolique, que la class untradifférentiable soit quasi-analytique ou non ; on obtient alors, des exemples d’idéaux fermés dans ${{C}_{M}}\left( {{\mathbb{R}}^{n}} \right)$ . On complète ce travail par une généralisation d’un résultat de C. L. Childress dans le cadre quasi-analytique et quelques remarques.

The cohomology groups of some sequence algebras are considered. In particular, the amenability of the James space $J_p$ is investigated; this space was first studied by R. C. James and was shown to be a Banach algebra for the pointwise product. Then a full classification of the closed ideals in $J_p$ is given, extending a result of N. J. Laustsen. Finally, some related algebras $\alp$ are considered, and it is established when these Banach algebras are amenable.

Let $B$ be a unital commutative Banach algebra, and let $A$ be an arbitrary subalgebra of $B$. Suppose that an ideal $I\subset A$ consists of elements that are non-invertible in $B$. Then there exists an ideal $J$ in $A$ that: (i) contains $I$, (ii) also consists of elements non-invertible in $B$, and (iii) is of codimension one in $A$. This result is used in the study of a class of subspectra on $A$.

Let $m$ be a point of the maximal ideal space of ${{H}^{\infty }}$ with nontrivial Gleason part $P\left( m \right)$. If ${{L}_{m}}\,:\,\text{D}\,\to \,\text{P(m)}$ is the Hoffman map, we show that ${{H}^{\infty }}\,\circ \,{{L}_{m}}$ is a closed subalgebra of ${{H}^{\infty }}$. We characterize the points $m$ for which ${{L}_{m}}$ is a homeomorphism in terms of interpolating sequences, and we show that in this case ${{H}^{\infty }}\,\circ \,{{L}_{m}}$ coincides with ${{H}^{\infty }}$. Also, if ${{I}_{m}}$ is the ideal of functions in ${{H}^{\infty }}$ that identically vanish on $P\left( m \right)$, we estimate the distance of any $f\,\in \,{{H}^{\infty }}\,\text{to}\,{{I}_{m}}$.

Let 𝒳 be a complex Banach space, with open unit ball B. We consider the algebra of analytic functions on B that are weakly continuous and that are uniformly continuous with respect to the norm. We show these are precisely the analytic functions on B that extend to be weak-star continuous on the closed unit ball of 𝒳**. If 𝒳* has the approximation property, then any such function is approximable uniformly on B by finite polynomials in elements of 𝒳*. On the other hand, there exist Banach spaces for which these finite-type polynomials fail to approximate. We consider also the approximation of entire functions by finite-type polynomials. Assuming 𝒳* has the approximation property, we show that entire functions are approximable uniformly on bounded sets if and only if the spectrum of the algebra of entire functions coincides (as a point set) with 𝒳**.

Extreme points of positive functionals and spectral states on real commutative Banach algebras are investigated and characterized as multiplicative functionals extending the well-known results from complex to real Banach algebras. As an application a new and short proof of the existence of the Shilov boundary of a real commutative Banach algebra with nonempty maximal ideal space is given.

In this paper we show that a ring derivation on a function algebra is trivial provided that the Choquet boundary of the algebra contains a dense sequentially non-isolated set.

We give a characterization of a sparse Blaschke product b in terms of the separation of support sets of its zeros in M(H∞ + C) and the structure of the nonanalytic points. We use this characterization to give a sufficient condition on an interpolating Blaschke product q to have the following property: there exists a non trivial Gleason part P on which q is nonzero and less than one.