Daniel Suárez
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 $F|_{P(m)} = f$.
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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 $H^2(M(H^\infty),\mathbb Z)=0$.
In this paper we study the maximal ideals in a commutative ring of bicomplex numbers and then we describe the maximal ideals in a bicomplex algebra. We found that the kernel of a nonzero multiplicative BC-linear functional in a commutative bicomplex Banach algebra need not be a maximal ideal. Finally, we introduce the notion of bicomplex division algebra and generalize the Gelfand-Mazur theorem for the bicomplex division Banach algebra.
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 $p\in(1,\infty)$.
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.
We prove the generalized Wolff's Ideal Theorem on certain uniformly closed subalgebras of $ H^{\infty}(\mathbb{D}) $ on which the Corona Theorem is already known to hold.
The unique maximal ideal in the Banach algebra $L(E)$, $E = (\oplus \ell^\infty(n))_{\ell^1}$, is identified. The proof relies on techniques developed by Laustsen, Loy and Read and a dichotomy result for operators mapping into $L^1$ due to Laustsen, Odell, Schlumprecht and Zsák.
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。
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…
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.
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.
We generalize the notion of $M$-ideals in order smooth $\infty$-normed spaces to "smooth $p$-order ideals" in order smooth $p$-normed spaces. We show that if $V$ is an order smooth $p$-normed space and $W$ is a closed subspace of $V$, then $W$ is a smooth $p$-order ideal in $V$ if and only if $W^{\perp}$ is a smooth $p'$-order ideal in order smooth $p'$-normed space if and only if $W^{\perp\perp}$ is a smooth $p$-order ideal in order smooth $p$-normed space $V^{**}$. We prove that every $L$-summand in order smooth $1$-normed space is a smooth $1$-order ideal. We find a condition under which ev…
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…
The monograph contains the detailed exposition of the results obtained by the author during the last several years. In particular it contains an improvement of the well known Calderon - Ryff interpolation theorem and description of "verifying" operators of conditional mathematical expectations {if such an operator leaves an order ideal in $L^1% invariant, then the ideal is an interpolation space}.
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…
Let $λ$ be an infinite cardinal number and let $\ell_\infty^c(λ)$ denote the subspace of $\ell_\infty(λ)$ consisting of all functions that assume at most countably many non-zero values. We classify all infinite dimensional complemented subspaces of $\ell_\infty^c(λ)$, proving that they are isomorphic to $\ell_\infty^c(κ)$ for some cardinal number $κ$. Then we show that the Banach algebra of all bounded linear operators on $\ell_\infty^c(λ)$ or $\ell_\infty(λ)$ has the unique maximal ideal consisting of operators through which the identity operator does not factor. Using similar techniques, we …
Using upper $\ell_p$-estimates for normalized weakly null sequence images, we describe a new family of operator ideals $\mathcal{WD}_{\ell_p}^{(\infty,ξ)}$ with parameters $1\leq p\leq\infty$ and $1\leqξ\leqω_1$. These classes contain the completely continuous operators, and are distinct for all choices $1\leq p\leq\infty$ and, when $p\neq 1$, for all choices $ξ\neqω_1$. For the case $ξ=1$, there exists an ideal norm $\|\cdot\|_{(p,1)}$ on the class $\mathcal{WD}_{\ell_p}^{(\infty,1)}$ under which it forms a Banach ideal.
What is an adequate extension of an operator ideal I to the polynomial and multilinear settings? This question motivated the appearance of the concepts of coherent sequences of polynomial ideals and compatibility of a polynomial ideal with an operator ideal, introduced by D. Carando el al. We propose a different approach by considering pairs (U_{k},M_{k})_{k=1}^{\infty}, where (U_{k})_{k=1}^{\infty} is a polynomial ideal and (M_{k})_{k=1}^{\infty} is a multi-ideal, instead of considering just polynomial ideals. Our approach ends a discomfort caused by the previous theory: for real scalars the …
Topology of the Maximal Ideal Space of $H^{\infty}$
研究了开单位圆盘$\Di\subset\Co$上有界解析函数代数$H^{\infty}=H^{\infty}(\Di)$的极大理想空间$M(H^{\infty})$的结构.基于$dim\ M(H^{\infty})=2$的事实，我们证明了$H^{\infty}$的矩阵值电晕定理。我们的结果在很大程度上依赖于描述定义在紧致流形的覆盖空间上的某些连续函数代数的极大理想空间的拓扑结构。
We study the lattice of closed ideals in the algebra of continuous linear operators acting on $p$th Tandori and $p'$th Cesàro sequence spaces, $1\leqslant p<\infty$, which we show are isomorphic to the classical sequence spaces $(\oplus_{n=1}^\infty\ell_\infty^n)_p$ and $(\oplus_{n=1}^\infty\ell_1^n)_{p'}$, respectively. We also show that Tandori sequence spaces are complemented in certain Lorentz sequence spaces, and that the lattice of closed ideals for certain other Lorentz and Garling sequence spaces has infinite cardinality.
We show that each maximal ideal in a commutative Banach algebra has codimension 1.
In this paper we give the complete characterization of the boundedness of the generalized fractional maximal operator $$ M_{ϕ,Λ^α(b)}f(x) : = \sup_{Q \ni x} \frac{\|f χ_Q\|_{Λ^α(b)}}{ϕ(|Q|)} \qquad (x \in {\mathbb R}^n), $$ between the classical Lorentz spaces $Λ^p (v)$ and $Λ^q(w)$ for appropriate functions $ϕ$, where $0 < p,\,q < \infty$, $0 < α\le r < \infty$, $v,w,\,b$ are weight functions on $(0,\infty)$ such that $0 < B(x): = \int_0^x b < \infty$, $x > 0$, $B \in Δ_2$ and $B(t) / t^{α/ r}$ is quasi-increasing.
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.
In this paper we study sharp pointwise inequalities for maximal operators. In particular, we strengthen DeVore's inequality for the moduli of smoothness and a logarithmic variant of Bennett--DeVore--Sharpley's inequality for rearrangements. As a consequence, we improve the classical Stein--Zygmund embedding deriving $\dot{B}^{d/p}_\infty L_{p,\infty}(\mathbb{R}^d) \hookrightarrow \text{BMO}(\mathbb{R}^d)$ for $1 < p < \infty$. Moreover, these results are also applied to establish new Fefferman--Stein inequalities, Calderón--Scott type inequalities, and extrapolation estimates. Our approach is …
The maximal operator associated to a non-symmetric Ornstein-Uhlenbeck semigroup
设（H_t）是R^d上具有协方差阵I和漂移矩阵λ（R-I）的Ornstein-Uhlenbeck半群，其中λ>0，R是斜伴随矩阵，用γ_ \infty表示（H_t）的不变测度.这种形式的半群是L^2（γ_ \infty）上正规的Ornstein-Uhlenbeck半群的基本构造块。我们证明了如果矩阵R生成一个单参数周期旋转群，则与该半群相关联的极大算子关于不变测度是弱1型的。证明了任意正规Ornstein-Uhlenbeck半群的极大算子在L^p（γ_ \infty）上有界当且仅当1<\le \infty .
Hilbert transform associated with finite maximal subdiagonal algebras
设$M$是具有忠实正规有限迹$t$的冯诺依曼代数，$H^\infty$是$M$的有限极大次对角线.证明了弱 * Dirichlet代数的共轭函数是$L^p(M,t)$到$L^p(M,t)$的有界线性映射，$1<p<\infty$是$L^1(M,t)$到$L^{1,\infty}(M,t)$的连续线性映射.我们还得到，如果正算子$a$使得$a\log^{+}a \in L^1(M,t)$，则它的共轭属于$L^1(M,t)$ .
Contractibility of Maximal Ideal Spaces of Certain Algebras of Almost Periodic Functions
研究了一类殆周期函数代数的极大理想空间的拓扑性质。我们的主要结果是，这样的空间是可收缩的。我们提出这个结果的某些推论。
The maximal operator on variable Lebesgue spaces: an ${\mathcal A}_{\infty}$-characterization
本文通过可变指数$A_{\infty}$条件，给出了最大算子在可变指数空间上有界的新的判据。
We study the ideal membership problem in $H^\infty$ on the unit disc. Thus, given functions $f,f_1,\ldots,f_n$ in $H^\infty$, we seek sufficient conditions on the size of $f$ in order for $f$ to belong to the ideal of $H^\infty$ generated by $f_1,\ldots,f_n$. We provide a different proof of a theorem of Treil, which gives the sharpest known sufficient condition. To this end, we solve a closely related problem in the Hilbert space $H^2$, which is equivalent to the ideal membership problem by the Nevanlinna-Pick property of $H^2$.
We address the following two questions regarding the maximal left ideals of the Banach algebra $\mathscr{B}(E)$ of bounded operators acting on an infinite-dimensional Banach pace $E$: (Q1) Does $\mathscr{B}(E)$ always contain a maximal left ideal which is not finitely generated? (Q2) Is every finitely-generated, maximal left ideal of $\mathscr{B}(E)$ necessarily of the form \{T\in\mathscr{B}(E): Tx = 0\} (*) for some non-zero $x\in E$? Since the two-sided ideal $\mathscr{F}(E)$ of finite-rank operators is not contained in any of the maximal left ideals given by (*), a positive answer to the se…
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。
Prüfer's Ideal Numbers as Gelfand's maximal Ideals
多元算术是与$p$ --进理论有关的数学分支。本文的目的是说明多进算术与交换巴拿赫代数的经典理论之间有着非常密切的关系。即，设$\ms A$是由$\Z$上所有具有一致范数的复周期函数组成的代数。多进拓扑环可以定义为所有特征$\ms A\to\C$都具有卷积运算和Gelfand拓扑的环.
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 …
On maximal ideals in certain reduced twisted C*-crossed products
本文研究离散群在含单位元C*-代数上的扭曲作用，给出了使得代数的极大不变理想与对应约化扭曲C*-交叉积的极大理想成双射对应的条件，推进了相关算子代数结构的研究。
In this paper we give precise characterizations of the relation between the Nevanlinna counting function and pull-back measure of an analytic self-map of the unit disk near the boundary. We show that it is quite worth considering these two concepts on the maximal ideal space of the bounded analytic functions.
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 …
Analytic primes, $M$-ideals, and $p$-sets in $H^\infty(\mathbb{D})$
该文研究H^∞(𝔻)中p-集、M-理想与解析素元的结构，证明H^∞(𝔻)的M-理想是解析素元且在哈代空间稠密，外函数起关键作用，部分结果适用于多圆盘，为H^∞(𝔻)提供新视角。
Maximal ideal spaces of invariant function algebras on compact groups
本文给出了紧群$G$上双不变函数代数$A$的极大理想空间（谱）$\cM_A$的刻划. $\cM_A$具有自然相容结构：它是一个具有对合、极式分解和解析结构的紧拓扑半群.给出了$\cM_A$的一个描述，并给出了函数代数的相关结果，例如，连通紧李群上的双不变函数代数是反对称的当且仅当其极大环面的Haar测度在其上是乘法的，其中一些结果被推广到了紧交换齐性空间的情形.由此，我们得到了紧致连通李群的交换（作为齐性空间）轨道的Hilbert-Mumford准则的无限维版本，其中一个参数半群被它们的有限序列所取代.关键词：极大理想空间，不变函数代数，复李半群，交换齐性空间，希尔伯特-芒福德准则
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…
Let $(S, d, ρ)$ be the affine group $\mathrm{R}^n \ltimes \mathrm{R}^+$ endowed with the left-invariant Riemannian metric $d$ and the right Haar measure $ρ$, which is of exponential growth at infinity. In this paper, for any linear operator $T$ on $(S, d, ρ)$ associated with a kernel $K$ satisfying certain integral size condition and Hörmander's condition, the authors prove that the following four statements regarding the corresponding maximal singular integral $T^\ast$ are equivalent: $T^\ast$ is bounded from $L_c^\infty$ to $\mathrm{BMO}$, $T^\ast$ is bounded on $L^p$ for all $p\in(1, \infty…
Weighted Lorentz spaces: sharp mixed $A_p-A_{\infty}$ estimate for maximal functions
本文在加权 Lorentz 空间中证明 Hardy–Littlewood 极大函数的精确混合 $A_p-A_{\infty}$ 估计，方法无需重排，并推广至双权、强极大算子及对偶情形。
The problem of ideals of $H^\infty$: beyond the exponent 3/2
本文讨论了$H^\infty$的理想问题：描述增函数$ϕ\ge 0$，使得对于单位圆$D$中的所有有界解析函数$f_1,f_2,...,f_n, τ$，条件$|τ(z) | \le ϕ(\sum_k |f_k(z)|)$对所有$z\in D$，意味着$τ$属于由$f_1,f_2,...,f_n$生成的理想.作者在前面已经证明了函数$ϕ(s) =s^2$不起作用。本文的主要结果是：$ϕ$可以取任何形式为$ϕ(s) =s^2 ψ(\ln s^{-2})$的函数，其中$ψ$是$[0, \infty)$上满足$\int_0^\infty ψ(x) dx <\infty$的有界非增函数.
A theorem of Giesy and James states that $c_0$ is finitely representable in James' quasi-reflexive Banach space $J_2$. We extend this theorem to the $p$th quasi-reflexive James space $J_p$ for each $p \in (1,\infty)$. As an application, we obtain a new closed ideal of operators on $J_p$, namely the closure of the set of operators that factor through the complemented subspace $(\ell_\infty^1 \oplus \ell_\infty^2 \oplus...\oplus \ell_\infty^n \oplus...)_{\ell_p}$ of $J_p$.
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.
Let $(\cx,\,d,\,μ)$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrically doubling condition. In this paper, the authors show that for the maximal Calderón-Zygmund operator associated with a singular integral whose kernel satisfies the standard size condition and the Hörmander condition, its $L^p(μ)$ boundedness with $p\in(1,\infty)$ is equivalent to its boundedness from $L^1(μ)$ into $L^{1,\infty}(μ)$. Moreover, applying this, together with a new Cotlar type inequality, the authors show that if the Calderón-Zygmund operator $T$ is bounded on $L^2(μ…
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。
We investigate certain ideals (associated with Blaschke products) of the analytic Lipschitz algebra $A^α$, with $α>1$, that fail to be "ideal spaces". The latter means that the ideals in question are not describable by any size condition on the function's modulus. In the case where $α=n$ is an integer, we study this phenomenon for the algebra $H^\infty_n=\{f:f^{(n)}\in H^\infty\}$ rather than for its more manageable Zygmund-type version. This part is based on a new theorem concerning the canonical factorization in $H^\infty_n$.
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。
Complemented ideals of $\ell_\infty$
本文研究了$\ell_\infty$中补理想的问题，通过分析Stone空间的可近似性，确定了满足条件的理想结构。
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