Contractibility of Maximal Ideal Spaces of Certain Algebras of Almost Periodic Functions

Extending polynomials in maximal and minimal ideals

Given an homogeneous polynomial on a Banach space $E$ belonging to some maximal or minimal polynomial ideal, we consider its iterated extension to an ultrapower of $E$ and prove that this extension remains in the ideal and has the same ideal norm. As a consequence, we show that the Aron-Berner extension is a well defined isometry for any maximal or minimal ideal of homogeneous polynomials. This allow us to obtain symmetric versions of some basic results of the metric theory of tensor products.

单位交换Banach代数中的非极大闭素理想是可及的

Non-maximal closed prime ideals in a unital commutative Banach algebra are accessible

本文证明了单位交换Banach代数中非极大闭素理想可表示为所有真包含它的闭理想之交，并由此推出半素情形下导子与满同态连续，以及分离理想必为幂零诣零理想。

On the maximal ideal space of even quasicontinuous functions on the unit circle

Let $PQC$ stand for the set of all piecewise quasicontionus function on the unit circle, i.e., the smallest closed subalgebra of $L^\infty(\mathbb{T})$ which contains the classes of all piecewise continuous function $PC$ and all quasicontinuous functions $QC=(C+H^\infty)\cap(C+\overline{H^\infty})$. We analyze the fibers of the maximal ideal spaces $M(PQC)$ and $M(QC)$ over maximal ideals from $M(\widetilde{QC})$, where $\widetilde{QC}$ stands for the $C^*$-algebra of all even quasicontinous functions. The maximal ideal space $M(\widetilde{QC})$ is decribed and partitioned into various subsets…

Maximal ideals in commutative Banach algebras

We show that each maximal ideal in a commutative Banach algebra has codimension 1.

Maximal ideals of generalized summing linear operators

We prove when a Banach ideal of linear operators defined, or characterized, by the transformation of vector-valued sequences is maximal. Known results are recovered as particular cases and new information is obtained. To accomplish this task we study a tensor quasi-norm determined by the underlying sequence classes. The duality theory for these tensor quasi-norms is also developed.

New characterizations of maximal ideals in algebras of continuous vector-valued functions

Let X be a compact Hausdorf space, let A be a commutative unital Banach algebra, and let C(X,A) denote the algebra of continuous A-valued functions on $X$ equipped with the uniform norm ||f||=sup{||f(x)||:x\in X} for all f in C(X,A). Hausner, in [Proc. Amer. Math. Soc. 8(1957), 246--249], proved that M is a maximal ideal in C(X,A) if and only if there exist a point x in X and a maximal ideal N in A such that M={f in C(X,A) : f(x) in N}. In this note, we give new characterizations of maximal ideals in C(X,A). We also present a short proof of Hausner's result by a different approach.

Generators of maximal left ideals in Banach algebras

In 1971, Grauert and Remmert proved that a commutative, complex, Noetherian Banach algebra is necessarily finite-dimensional. More precisely, they proved that a commutative, complex Banach algebra has finite dimension over $\C$ whenever all the closed ideals in the algebra are (algebraically) finitely generated. In 1974, Sinclair and Tullo obtained a non-commutative version of this result. In 1978, Ferreira and Tomassini improved the result of Grauert and Remmert by showing that the statement is also true if one replaces `closed ideals' by `maximal ideals in the Šilov boundary of $A$'. We shal…

Uniqueness of the maximal ideal of the Banach algebra of bounded operators on $C([0,ω_1])$

Let $ω_1$ be the first uncountable ordinal. By a result of Rudin, bounded operators on the Banach space $C([0,ω_1])$ have a natural representation as $[0,ω_1]\times 0,ω_1]$-matrices. Loy and Willis observed that the set of operators whose final column is continuous when viewed as a scalar-valued function on $[0,ω_1]$ defines a maximal ideal of codimension one in the Banach algebra $\mathscr{B}(C([0,ω_1]))$ of bounded operators on $C([0,ω_1])$. We give a coordinate-free characterization of this ideal and deduce from it that $\mathscr{B}(C([0,ω_1]))$ contains no other maximal ideals. We then obt…

On the analytic structure of the $H^\infty$ maximal ideal space

We characterize the algebra $H^\infty \circ L_{m}$, where $m$ is a point of the maximal ideal space of $H^\infty$ with nontrivial Gleason part $P(m)$ and $L_{m} : \mathbb{D}\to P(m)$ is the coordinate Hoffman map. In particular, it is shown that for any continuous function $f: P(m) \to \mathbb{C}$ with $f\circ L_{m} \in H^\infty$ there exists $F\in H^\infty$ such that $F|_{P(m)} = f$.

Topology of the Maximal Ideal Space of $H^{\infty}$

We study the structure of the maximal ideal space $M(H^{\infty})$ of the algebra $H^{\infty}=H^{\infty}(\Di)$ of bounded analytic functions defined on the open unit disk $\Di\subset\Co$. Based on the fact that $dim\ M(H^{\infty})=2$ we prove for $H^{\infty}$ the matrix-valued corona theorem. Our results heavily rely on the topological construction describing maximal ideal spaces of certain algebras of continuous functions defined on the covering spaces of compact manifolds.

Topology of the Maximal Ideal Space of $H^\infty$ Revisited

Let $M(H^\infty)$ be the maximal ideal space of the Banach algebra $H^\infty$ of bounded holomorphic functions on the unit disk $\mathbb D\subset\mathbb C$. We prove that $M(H^\infty)$ is homeomorphic to the Freudenthal compactification $γ(M_a)$ of the set $M_a$ of all non-trivial (analytic disks) Gleason parts of $M(H^\infty)$. Also, we give alternative proofs of important results of Suárez asserting that the set $M_s$ of trivial (one-pointed) Gleason parts of $M(H^\infty)$ is totally disconnected and that the Čech cohomology group $H^2(M(H^\infty),\mathbb Z)=0$.

Uniqueness of the maximal ideal of operators on the $\ell_p$-sum of $\ell_\infty^n\ (n\in\mathbb{N})$ for $1<p<\infty$

A recent result of Leung (Proceedings of the American Mathematical Society, to appear) states that the Banach algebra $\mathscr{B}(X)$ of bounded, linear operators on the Banach space $X=\bigl(\bigoplus_{n\in\mathbb{N}}\ell_\infty^n\bigr)_{\ell_1}$ contains a unique maximal ideal. We show that the same conclusion holds true for the Banach spaces $X=\bigl(\bigoplus_{n\in\mathbb{N}}\ell_\infty^n\bigr)_{\ell_p}$ and $X=\bigl(\bigoplus_{n\in\mathbb{N}}\ell_1^n\bigr)_{\ell_p}$ whenever $p\in(1,\infty)$.

强截断与极大理想原理

Strong truncations and Maximal Ideal Principles

本文比较了两种极大理想存在原理，一种针对具有强单位的向量格，另一种新引入的针对具有强截断的向量格，并证明在ZF集合论下两者等价。

Oka Principle on the Maximal Ideal Space of ${\mathbf H^\infty}$

The classical Grauert and Ramspott theorems constitute the foundation of the Oka principle on Stein spaces. In this paper we establish analogous results on the maximal ideal space $M(H^\infty)$ of the Banach algebra $H^\infty$ of bounded holomorphic functions on the open unit disk $\mathbb D\subset\mathbb C$. We illustrate our results by some examples and applications to the theory of operator-valued $H^\infty$ functions.

Prüfer's Ideal Numbers as Gelfand's maximal Ideals

Polyadic arithmetics is a branch of mathematics related to $p$--adic theory. The aim of the present paper is to show that there are very close relations between polyadic arithmetics and the classic theory of commutative Banach algebras. Namely, let $\ms A$ be the algebra consisting of all complex periodic functions on $\Z$ with the uniform norm. Then the polyadic topological ring can be defined as the ring of all characters $\ms A\to\C$ with convolution operations and the Gelfand topology.

巴拿赫空间上有界算子构成的巴拿赫代数的极大右理想

Maximal right ideals of the Banach algebra of bounded operators on a Banach space

该研究探讨巴拿赫代数$\boldsymbol{\beta}(E)$的极大右理想，明确有限生成提升理想的算子论障碍，通过相关方法对多类巴拿赫空间构造非有限生成极大右理想，还确定部分空间的有限生成极大右理想均为固定的。

Topology of Gleason Parts in maximal ideal spaces with no analytic discs

We strengthen, in various directions, the theorem of Garnett that every $σ$-compact, completely regular space $X$ occurs as a Gleason part for some uniform algebra. In particular, we show that the uniform algebra can always be chosen so that its maximal ideal space contains no analytic discs. We show that when the space $X$ is metrizable, the uniform algebra can be chosen so that its maximal ideal space is metrizable as well. We also show that for every locally compact subspace $X$ of a Euclidean space, there is a compact set $K$ in some ${\mathbb C}^N$ so that $\hat K \setminus K$ contains a …

Maximal ideal spaces of invariant function algebras on compact groups

The paper contains a description of the maximal ideal spaces (spectra) $\cM_A$ of bi-invariant function algebras $A$ on a compact group $G$. There are natural compatible structures in $\cM_A$: it is a compact topological semigroup with involution, polar decomposition, and analytic structure. The paper contains a description of $\cM_A$ and related results on function algebras; for example, a bi-invariant function algebra on a connected compact Lie group is antisymmetric if and only if the Haar measure of its maximal torus is multiplicative on it. Some results are extended to the case of compact…

Maximal ideals in rings of real measurable functions

Let $ M (X)$ be the ring of all real measurable functions on a measurable space $(X, \mathscr{A})$. In this article, we show that every ideal of $M(X)$ is a $Z^{\circ}$-ideal. Also, we give several characterizations of maximal ideals of $M(X)$, mostly in terms of certain lattice-theoretic properties of $\mathscr{A}$. The notion of $T$-measurable space is introduced. Next, we show that for every measurable space $(X,\mathscr{A})$ there exists a $T$-measurable space $(Y,\mathscr{A}^{\prime})$ such that $M(X)\cong M(Y)$ as rings. The notion of compact measurable space is introduced. Next, we prov…

Factorisation in stopping-time Banach spaces: identifying unique maximal ideals

Stopping-time Banach spaces is a collective term for the class of spaces of eventually null integrable processes that are defined in terms of the behaviour of the stopping times with respect to some fixed filtration. From the point of view of Banach space theory, these spaces in many regards resemble the classical spaces such as $L^1$ or $C(Δ)$, but unlike these, they do have unconditional bases. In the present paper we study the canonical bases in the stopping-time spaces in relation to factorising the identity operator thereon. Since we work exclusively with the dyadic-tree filtration, this …

Asymptotic ideals (ideals in the ring of Colombeau generalized constants with continuous parametrization)

We study the asymptotics at zero of continuous functions on (0, 1] by means of their asymptotic ideals, i.e., ideals in the ring of continuous functions on (0, 1] satisfying a polynomial growth condition at 0 modulo rapidly decreasing functions at 0. As our main result, we characterize maximal and prime ideals in terms of maximal and prime filters.

具有可数无点像的函数环中的极大理想

Maximal Ideals in Functions Rings with a Countable Pointfree Image

本文研究定义在框架上的具有可数无点像的连续实值函数子环，证明其为 clean 环，并利用 ${\bf M}_c$-理想刻画其极大理想，建立与框架点及各类紧化之间的对应关系。

关于 $\mathcal I$-null 序列空间的补性

On the complementation of spaces of $\mathcal I$-null sequences

本文研究了$c_{0,\mathcal{I}}$空间在$\ell_\infty$中的补性，证明其与理想是否为$\omega$-极大理想相关，并刻画了商空间为有限维的条件。

非平凡统一代数在其最大理想空间上狄利克雷

A nontrivial uniform algebra Dirichlet on its maximal ideal space

本文构造了一个非平凡统一代数，证明其在最大理想空间上为狄利克雷，并解决了韦尔默和达勒等人提出的两个长期悬而未决的问题。

A characterization of maximal ideals in the Fréchet algebras of holomorphic functions $F^p$ $(1<p<\infty$

The space $F^p$ ($1<p<\infty$) consists of all holomorphic functions $f$ on the open unit disk $\Bbb D$ for which $\lim_{r\to 1}(1-r)^{1/q}\log^+M_{\infty}(r,f)=0,$ where $M_{\infty}(r,f)=\max_{\vert z\vert\le r}\vert f(z)\vert$ with $0<r<1$. Stoll [5, Theorem 3.2] proved that the space $F^p$ with the topology given by the family of seminorms $\left\{\Vert \cdot\Vert_{q,c}\right\}_{c>0}$ defined for $f\in F^q$ as $\Vert f\Vert_{q,c}:=\sum_{n=0}^{\infty}\vert a_n\vert\exp\left(-cn^{1/(q+1)} \right)<\infty,$ is a countably normed Fréchet algebra. Notice that for each $p>1$, $F^p$ is the Fréchet …

The ideal structure of measure algebras and asymptotic properties of group representations

We classify the weak*-closed maximal left ideals of the measure algebra $M(G)$ for certain Hermitian locally compact groups $G$ in terms of the irreducible representations of $G$ and their asymptotic properties. In particular, we obtain a classification for connected nilpotent Lie groups, and the Euclidean rigid motion groups. We also prove a version of this result for certain weighted measure algebras. We apply our classification to obtain an analogue of Barnes' Theorem on integrable representations for representations vanishing at infinity. We next study the relationship between weak*-closed…

A Short Note on some open problems in the geometry of operator ideals

We list and discuss the background of some open problems, regarding the principle of local reflexivity for maximal Banach ideals.