Nigel J. Kalton, Dirk Werner
Let $1<p,\,q<\infty$. It is shown for complex scalars that there are no nontrivial $M$-ideals in $L(L_p[0,1])$ if $p\neq 2$, and $K(\ell_p(\ell_q^n)$ is the only nontrivial $M$-ideal in $L(\ell_p(\ell_q^n)$. This proves a conjecture of C.-M. Cho and W. B. Johnson.
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In this article, we introduce the Lipschitz bounded approximation property for operator ideals. With this notion, we extend the original work done by Godefroy and Kalton and give some partial answers on the equivalence between the bounded approximation property and the Lipschitz bounded approximation property based on an arbitrary operator ideal. Furthermore, we investigate the three space problems on the preceding bounded approximation properties.
For a Banach algebra A and a closed ideal I, the notion of bounded approximate diagonal modulo I has been studied and investigated. In this paper we define the notion of locally bounded approximate diagonal modulo an ideal I for a Frechet algebra A and obtain the relation between amenability modulo an ideal I and the existence of locally bounded approximate diagonals modulo I.
We prove that in the reflexive range $1<p<q<\infty$ the algebra of all bounded linear operators on $\ell_p\oplus\ell_q$ has infinitely many closed ideals. This solves a problem raised by A. Pietsch in his book `Operator ideals'.
Suppose that $X$ is a Banach space. We will show that $X$ does not contain a copy of $c_0$ if and only if for each series which is not unconditionally convergent in $X$ respective sets coding all bounded subseries and rearrangements are meager. We use Bessaga-Pełczy{ń}ski $c_0-$Theorem and concept of uniformly unconditionally bounded series. Moreover we prove similar result for the ideal boundedness for a class of Baire ideals using Talagrand's characterisation.
We classify the closed ideals of bounded operators acting on the Banach spaces $\left(\bigoplus_{n \in \mathbb{N}} \ell_2^n\right)_{c_0} \oplus c_0(Γ)$ and $\left(\bigoplus_{n \in \mathbb{N}} \ell_2^n\right)_{\ell_1} \oplus \ell_1(Γ)$ for every uncountable cardinal $Γ$.
Functions of bounded variation from ideal perspective
本文利用自然数集上的理想理论，统一处理Waterman空间和Chanturia类这两类由变差定义的巴拿赫空间。
It is shown that a locally compact group $G$ is amenable if and only if some certain closed ideals of the Figà-Talamanca-Herz algebra $A_{p}(G)$ admit bounded $Δ$-weak approximate identities. Also, similar results are obtained for the function algebras $\mathfrak{L}A(G)$ and $C_{0}^{w}(G)$.
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…
Compactness in spaces of functions of bounded variation from ideal perspective
本文基于理想理论，给出了不同Waterman空间之间及不同Chanturia类之间紧嵌入的完整刻画，分别从定义序列和对应理想的性质两个角度进行描述。
We prove that the Berkovich space of the algebra of bounded analytic functions on the open unit disk of an algebraically closed nonarchimedean field contains multiplicative seminorms that are not norms and whose kernel is not a maximal ideal. We also prove that in general these seminorms are not univocally determined by their kernels, and provide a method for obtaining families of different seminorms sharing the same kernel. On the other hand, we prove that there are also kernels that cannot be obtained by that method. The relation with the Berkovich space of the Tate algebra is also given.
Amenability modulo an ideal of a Banach algebra have been defined and studied. In this paper we introduce the concept of amenability modulo an ideal of a Frechet algebra and investigate some known results about amenability modulo an ideal of a Frechet algebra. Also we show that a Frechet algebra A is amenable modulo an ideal if and only if A is isomorphic to a reduced limit of amenable modulo an ideal of Banach algebras.
In this paper we give an example of a Banach algebra $A$ and a closed ideal $I$ of $A$ such that the multiplier algebra of $I$ is equal to $A$ but $I$ does not have any bounded approximate identity. In the case that $I$ has an approximate identity, we give a necessary condition on $I$ for which $A=\mathcal{M}(I)$, where $\mathcal{M}(I)$ denotes the multiplier algebra of $I$. Finally, as a corollary of our results, we show that the Fourier algebra of an amenable group is strictly dense in the Fourier-Stieltjes algebra.
Fixed a Banach operator ideal $\mathcal A$, we introduce and investigate two new approximation properties, which are strictly weaker than the bounded approximation property (BAP) for $\mathcal A$ of Lima, Lima and Oja (2010). We call them the weak BAP for $\mathcal A$ and the local BAP for $\mathcal A$, showing that the latter is in turn strictly weaker than the former. Under this framework, we address the question of approximation properties passing from dual spaces to underlying spaces. We relate the weak and local BAPs for $\mathcal A$ with approximation properties given by tensor norms and…
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。
In this paper, we introduce and study some notations of amenability such as $n$-ideal amenability and $n$-weak amenability for Frechet algebra and we examine how these concepts in Banach algebra can be generalized and defined for Frechet algebra. We will also examine some inherited properties of $n$-ideal and $n$-weak amenable Frechet algebra and determine the relations between $m$ and $n$-weak amenability and $m$ and $n$-ideal amenability for Frechet algebra. Also, the relation of these new concepts of amenability of a Frechet algebra and its unitization is investigated.
Applying a linearization theorem due to J. Mujica, we study the ideals of bounded holomorphic mappings $\mathcal{H}^\infty\circ\mathcal{I}$ generated by composition with an operator ideal $\mathcal{I}$. The bounded-holomorphic dual ideal of $\mathcal{I}$ is introduced and its elements are characterized as those that admit a factorization through $\mathcal{I}^\mathrm{dual}$. For complex Banach spaces $E$ and $F$, we also analyze new ideals of bounded holomorphic mappings from an open subset $U\subseteq E$ to $F$ such as $p$-integral holomorphic mappings and $p$-nuclear holomorphic mappings with…
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.
Let A and B be two commutative semisimple Frechet algebras. We first give a characterization of the multiplier algebra of the direct sum of A and B. We then prove that A \oplus B is a BSE-algebra if and only if A and B are BSE-algebras. Furthermore, for a closed ideal I of A, we study multipliers of ideals and quotient algebras of A and show that I and A/I are BSE-algebras, under certain conditions.
This paper concerns the notion of a symmetric algebra and its generalization to a quasi-symmetric algebra. We study the structure of these algebras in respect to their hull-kernel regularity and existence of some ideals, especially the hull-minimal ideals.
In this paper we study the ideal amenability of Banach algebras. Let $\cal A$ be a Banach algebra and let $I$ be a closed two-sided ideal in $\cal A$, $\cal A$ is $I$-weakly amenable if $H^{1}({\cal A},I^*)=\{0\}$. Further, $\cal A$ is ideally amenable if $\cal A$ is $I$-weakly amenable for every closed two-sided ideal $I$ in $\cal A$. We know that a continuous homomorphic image of an amenable Banach algebra is again amenable. We show that for ideal amenability the homomorphism property for suitable direct summands is true similar to weak amenability and we apply this result for ideal amenabil…
We establish an analogue of Wolff's theorem on ideals in $H^{\infty}(\mathbb{D})$ for the multiplier algebra of Dirichlet space.
Let $\cal A$ be a Banach algebra. We study those closed ideals $I$ of $\cal A$ for which the first cohomology group of $\cal A$ with coefficients in $I^*$ is trivial; i.e. $H^1(\cal A,I^*)=\{0\}$. We investigate such closed ideals when $\cal A$ is weakly amenable or biflat. Also we give some hereditary properties of ideal amenability.
Let $G$ be a locally compact group. We prove that the augmentation ideal in $L^1(G)$ is (algebraically) finitely-generated as a left ideal if and only if $G$ is finite. We then investigate weighted versions of this result, as well as a version for semigroup algebras. Weighted measure algebras are also considered. We are motivated by a recent conjecture of Dales and Żelazko, which states that a unital Banach algebra in which every maximal left ideal is finitely-generated is necessarily finite-dimensional. We prove that this conjecture holds for many of the algebras considered. Finally, we use t…
It has been established by Inoue that a complex locally C*-algebra with a dense ideal posesses a bounded approximate identity which belonges to that ideal. It has been shown by Fritzsche that if a unital complex locally C*-algebra has an unbounded element then it also has a dense one-sided ideal. In the present paper we obtain analogues of the aforementioned results of Inoue and Fritzsche for real locally C*-algebras (projective limits of projective families of real C*-algebras), and for locally JB-algebras (projective limits of projective families of JB-algebras).
We show that each maximal ideal in a commutative Banach algebra has codimension 1.
We determine the ideal structure of the Toeplitz C*-algebra on the bidisk.
We continue the study of r-ideals, l-ideals, and HSA's in operator algebras. Some applications are made to the structure of operator algebras, including Wedderburn type theorems for a class of operator algebras. We also consider the one-sided $M$-ideal structure of certain tensor products of operator algebras.
A notion of partial ideal for an operator algebra is a weakening the notion of ideal where the defining algebraic conditions are enforced only in the commutative subalgebras. We show that, in a von Neumann algebra, the ultraweakly closed two-sided ideals, which we call total ideals, correspond to the unitarily invariant partial ideals. The result also admits an equivalent formulation in terms of central projections. We place this result in the context of an investigation into notions of spectrum of noncommutative $C^*$-algebras.
We construct a Banach space $Z$ such that the lattice of closed two-sided ideals of the Banach algebra $\mathscr{B}(Z)$ of bounded operators on $Z$ is as follows: $$ \{0\}\subset \mathscr{K}(Z)\subset\mathscr{E}(Z) \raisebox{-.5ex}% {\ensuremath{\overset{\begin{turn}{30}$\subset$\end{turn}}% {\begin{turn}{-30}$\subset$\end{turn}}}}\!\!% \begin{array}{c}\mathscr{M}_1\\[1mm]\mathscr{M}_2\end{array}\!\!\!% \raisebox{-1.25ex}% {\ensuremath{\overset{\raisebox{1.25ex}{\ensuremath{\begin{turn}{-30}$\subset$\end{turn}}}}% {\raisebox{-.25ex}{\ensuremath{\begin{turn}{30}$\subset$\end{turn}}}}}}\,\mathsc…
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…
We characterize the ideal of continuous-trace elements in a separable transformation-group $C^{*}$-algebra $C_0(X)\times G$. In addition, we identify the largest Fell ideal, the largest liminal ideal and the largest postliminal ideal.
Schur bounded patterns, submajorisation and operator Lipschitz functions
本文研究了非闭合子主要化的算子理想Schur bounded patterns，分析了Schatten理想C_p（0<p<1）的特性及非闭合子主要化的理想特征。
A $C^{*}$-algebra $A$ has ideal property if any ideal $I$ of $A$ is generated as a closed two sided ideal by the projections inside the ideal. Suppose that the limit $C^{*}$-algebra $A$ of inductive limit of direct sums of matrix algebras over spaces with uniformly bounded dimension has ideal property. In this paper, we will prove that $A$ can be written as an inductive limit of certain very special subhomogeneous algebras, namely, direct sum of dimension drop interval algebras and matrix algebras over 2-dimensional spaces with torsion $H^{2}$ groups.
We give the smallest closed ideal with given hull and inner factor for some weighted big Lipschitz algebras of analytic functions.
The notion of Fréchet locally $C^*$-algebra generalizes the notion of $C^*$-algebra. In this paper, we first present some definitions and basic facts about locally $C^*$-algebra, and then we introduce and study the notion of ideal and weak amenability for these algebras. Also, we show that every Fréchet locally $C^*$-algebra is ideally amenable.
A left ideal of any $C^*$-algebra is an example of an operator algebra with a right contractive approximate identity (r.c.a.i.). Indeed left ideals in $C^*$-algebras may be characterized as the class of such operator algebras, which happen also to be triple systems. Conversely, we show here and in a sequel to this paper, that operator algebras with r.c.a.i. should be studied in terms of a certain left ideal of a $C^*$-algebra. We study left ideals from the perspective of `Hamana theory' and using the multiplier algebras introduced by the author. More generally, we develop some general theory f…
A left ideal of any C*-algebra is an example of an operator algebra with a right contractive approximate identity (r.c.a.i.). Conversely, we show here and in a `pre-quel' to this paper [B], that operator algebras with r.c.a.i. should be studied in terms of a certain left ideal of a C*-algebra. We study operator algebras and their multiplier algebras from the perspective of `Hamana theory' and using the multiplier algebras introduced by the first author.
We study some topological properties of maximal ideal spaces of certain algebras of almost periodic functions. Our main result is that such spaces are contractible. We present certain corollaries of this result.
We show that a $C^\star$-algebra $B$ contains a dense left or right Fréchet ideal $A$, with $A$ a nuclear locally convex space, if and only if the primitive ideal space Prim$(B)$ of $B$ is discrete and countable, and $B/I$ is finite dimensional for each $I \in $ Prim$(B)$. We show the forward implication holds for a general Banach algebra $B$, if the ideal is assumed two-sided. For $C^\star$-algebras, we construct all two-sided dense nuclear ideals by defining a set of matrix-valued Schwartz functions on the countable discrete space Prim$(B)$.
For a locally compact Hausdorff space $X$ and a $C^*$-algebra $A$ with only finitely many closed ideals, we discuss a characterization of closed ideals of $C_0(X,A) $ in terms of closed ideals of $A$ and certain (compatible) closed subspaces of $X$. We further use this result to prove that a closed ideal of $C_0(X) \otimes^{\min} A$ is a finite sum of product ideals. We also establish that for a unital $C^*$-algebra $A$, $C_0(X,A)$ has centre-quotient property if and only if $A$ has centre-quotient property. As an application, we characterize the closed Lie ideals of $C_0(X,A)$ and identify al…
We obtain a complete description of closed ideals of the algebra $\mathcal{D}\cap \mathrm{lip}_α},$ $0<α\leq{1/2},$ where $\mathcal{D}$ is the Dirichlet space and $\mathrm{lip}_α}$ is the algebra of analytic functions satisfying the Lipschitz condition of order $α.$
In this paper, we study the rigidity of uniform Roe algebras via the ideal structures. We showed that for given metric spaces X and Y with bounded geometry, if their uniform Roe algebras are isomorphic, then they are coarse equivalent.
Let $\mathcal{I}\subseteq\mathcal{P}(ω)$ be a meager ideal. Then there are no continuous projections from $\ell_\infty$ onto the set of bounded sequences which are $\mathcal{I}$-convergent to $0$. In particular, it follows that the set of bounded sequences statistically convergent to $0$ is not isomorphic to $\ell_\infty$.
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