We show that there are $2^{2^{\aleph_0}}$ different closed ideals in the Banach algebra $L(L_p(0,1))$, $1<p\not= 2<\infty$. This solves a problem in A. Pietsch's 1978 book "Operator Ideals". The proof is quite different from other methods of producing closed ideals in the space of bounded operators on a Banach space; in particular, the ideals are not contained in the strictly singular operators and yet do not contain projections onto subspaces that are non Hilbertian. We give a criterion for a space with an unconditional basis to have $2^{2^{\aleph_0}}$ closed ideals in terms of the existence of a single operator on the space with some special asymptotic properties. We then show that for $1<q<2$ the space ${\frak X}_q$ of Rosenthal, which is isomorphic to a complemented subspace of $L_q(0,1)$, admits such an operator.
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…
It is well known that the only proper non-trivial norm-closed ideal in the algebra L(X) for X=\ell_p (1 \le p < \infty) or X=c_0 is the ideal of compact operators. The next natural question is to describe all closed ideals of L(\ell_p\oplus\ell_q) for 1 \le p,q < \infty, p \neq q, or, equivalently, the closed ideals in L(\ell_p,\ell_q) for p < q. This paper shows that for 1 < p < 2 < q < \infty there are at least four distinct proper closed ideals in L(\ell_p,\ell_q), including one that has not been studied before. The proofs use various methods from Banach space theory.
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.
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'.
Every bounded linear operator on a Hilbert space which is invertible modulo compact operators has a closed range and is, thus, generalized invertible. We consider the analogue question in general $C^*$-algebras and describe the closed ideals (called Moore-Penrose ideals in what follows) with the property that whenever an element is invertible modulo that ideal, then it is generalized invertible. In particular, we will see that the class of Moore-Penrose ideals coincides with the class of the dual ideals. Finally, we study some questions related with the projection lifting property of Moore-Pen…
We prove that the spaces $\mathcal L(\ell_p,\mathrm{c}_0)$, $\mathcal L(\ell_p,\ell_\infty)$ and $\mathcal L(\ell_1,\ell_q)$ of operators with $1<p,q<\infty$ have continuum many closed ideals. This extends and improves earlier works by Schlumprecht and Zsák, by Wallis, and by Sirotkin and Wallis. Several open problems remain. Key to our construction of closed ideals are matrices with the Restricted Isometry Property that come from Compressed Sensing.
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 $Γ$.
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.
In Archimedean vector lattices bands can be introduced via three different coinciding notions. First, they are order closed ideals. Second, they are precisely those ideals which equal their double disjoint complements. The third concept is that of an ideal which contains the supremum of any of its bounded subsets, provided the supremum exists in the vector lattice. We investigate these three notions and their relationships in the more general setting of Archimedean pre-Riesz spaces. We introduce the notion of a supremum closed ideal, which is related to the third aforementioned notion in vecto…
We construct various examples of non-trivial closed ideals of the compact-by-approximable algebra $\mathfrak{A}_X =:\mathcal K(X)/\mathcal A(X)$ on Banach spaces $X$ failing the approximation property. The examples include the following: (i) if $X$ has cotype $2$, $Y$ has type $2$, $\mathfrak{A}_X \neq \{0\}$ and $\mathfrak{A}_Y \neq \{0\}$, then $\mathfrak{A}_{X \oplus Y}$ has at least $2$ closed ideals, (ii) there are closed subspaces $X \subset \ell^p$ for $4 < p < \infty$ and $X \subset c_0$ such that $\mathfrak{A}_X$ contains a non-trivial closed ideal, (iii) there is a Banach space $Z$ s…
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 $C_{0}^{w}(G)$.
Theorem A and Theorem B of [1] state that for $1<p<\infty$ the lattice of closed ideals of $\mathcal{L}(\ell_p,c_0)$, $\mathcal{L}(\ell_p,\ell_\infty)$ and of $\mathcal{L}(\ell_1,\ell_p)$ are at least of cardinality $2^ω$. Here we show that the cardinality of the lattice of closed ideals of $\mathcal{L}(\ell_p,c_0)$, $\mathcal{L}(\ell_p,\ell_\infty)$ and of $\mathcal{L}(\ell_1,\ell_p)$, is at least $2^{2^ω}$, and thus equal to it.
We formulate general conditions which imply that $L(X,Y)$, the space of operators from a Banach space $X$ to a Banach space $Y$, has $2^{\mathfrak c}$ closed ideals where $\mathfrak c$ is the cardinality of the continuum. These results are applied to classical sequence spaces and Tsirelson type spaces. In particular, we prove that the cardinality of the set of closed ideals in $L(\ell_p\oplus\ell_q)$ is exactly $2^{\mathfrak c}$ for all $1<p<q<\infty$, which in turn gives an alternate proof of the recent result of Johnson and Schechtman that $L(L_p)$ also has $2^{\mathfrak c}$ closed ideals fo…
We obtain a complete description of closed ideals in weighted Lipschitz algebras $Λ_ω$ of analytic functions on the unit disk satisfying the following condition $$\frac{|f(z)-f(w)|}{ω(|z-w|)}=o(1)\qquad(as |z-w| \longrightarrow 0),$$ where $ω$ is a modulus of continuity satisfying some regularity conditions.
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 $α.$
Suppose $X$ is a real or complexified Banach space containing a complemented copy of $\ell_p$, $p\in(1,2)$, and a copy (not necessarily complemented) of either $\ell_q$, $q\in(p,\infty)$, or $c_0$. Then $\mathcal{L}(X)$ and $\mathcal{L}(X^*)$ each admit continuum many closed ideals. If in addition $q\geq p'$, $\frac{1}{p}+\frac{1}{p'}=1$, then the closed ideals of $\mathcal{L}(X)$ and $\mathcal{L}(X^*)$ each fail to be linearly ordered. We obtain additional results in the special cases of $\mathcal{L}(\ell_1\oplus\ell_q)$ and $\mathcal{L}(\ell_p\oplus c_0)$, $1<p<2<q<\infty$.
Let $\mathscr{B}(X)$ denote the Banach algebra of bounded operators on $X$, where~$X$ is either Tsirelson's Banach space or the Schreier space of order $n$ for some $n\in\mathbb N$. We show that the lattice of closed ideals of~$\mathscr{B}(X)$ has a very rich structure; in particular $\mathscr{B}(X)$ contains at least continuum many maximal ideals. Our approach is to study the closed ideals generated by the basis projections. Indeed, the unit vector basis is an unconditional basis for each of the above spaces, so there is a basis projection $P_N\in\mathscr{B}(X)$ corresponding to each non-empt…
The closed one-sided ideals of a C*-algebra are exactly the closed subspaces supported by the orthogonal complement of a closed projection. Let A be a (not necessarily selfadjoint) subalgebra of a unital C*-algebra B which contains the unit of B. Here we characterize the right ideals of A with left contractive approximate identity as those subspaces of A supported by the orthogonal complement of a closed projection in B** which also lies in the weak* closure of A. Although this seems quite natural, the proof requires a set of new techniques which may may be viewed as a noncommutative version o…
For a set $E\subset\mathbb{R}^n$ that contains the origin we consider $I^m(E)$ -- the set of all $m^{\text{th}}$ degree Taylor approximations (at the origin) of $C^m$ functions on $\mathbb{R}^n$ that vanish on $E$. This set is a proper ideal in $\mathcal{P}^m(\mathbb{R}^n)$ -- the ring of all $m^{\text{th}}$ degree Taylor approximations of $C^m$ functions on $\mathbb{R}^n$. In [FS] we introduced the notion of a \textit{closed} ideal in $\mathcal{P}^m(\mathbb{R}^n)$, and proved that any ideal of the form $I^m(E)$ is closed. In this paper we classify (up to a natural equivalence relation) all cl…
The main result is that there are infinitely many; in fact, a continuum; of closed ideals in the Banach algebra $L(L_1)$ of bounded linear operators on $L_1(0,1)$. This answers a question from A. Pietsch's 1978 book "Operator Ideals". The proof also shows that $L(C[0,1])$ contains a continuum of closed ideals. Finally, a duality argument yields that $L(\ell_\infty)$ has a continuum of closed ideals.
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 …
Let $Γ$ be the unit circle, $A(Γ)$ the Wiener algebra of continuous functions whose series of Fourier coefficients are absolutely convergent, and $A^+$ the subalgebra of $A(Γ)$ of functions whose negative coefficients are zero. If $I$ is a closed ideal of $A^+$, we denote by $S_I$ the greatest common divisor of the inner factors of the nonzero elements of $I$ and by $I^A$ the closed ideal generated by $I$ in $A(Γ)$. It was conjectured that the equality $I^A= S_I H^\infty \cap I^A$ holds for every closed ideal $I$. We exhibit a large class $\scr F$ of perfect subsets of $Γ$, including the triad…
We show that every Lorentz sequence space $d(\textbf{w},p)$ admits a 1-complemented subspace $Y$ distinct from $\ell_p$ and containing no isomorph of $d(\textbf{w},p)$. In the general case, this is only the second nontrivial complemented subspace in $d(\textbf{w},p)$ yet known. We also give an explicit representation of $Y$ in the special case $\textbf{w}=(n^{-θ})_{n=1}^\infty$ ($0<θ<1$) as the $\ell_p$-sum of finite-dimensional copies of $d(\textbf{w},p)$. As an application, we find a sixth distinct element in the lattice of closed ideals of $\mathcal{L}(d(\textbf{w},p))$, of which only five …
Let $X$ be a completely regular topological space. We study closed ideals $H$ of $C_B(X)$, the normed algebra of bounded continuous scalar-valued mappings on $X$ equipped with pointwise addition and multiplication and the supremum norm, which are non-vanishing, in the sense that, there is no point of $X$ at which every element of $H$ vanishes. This is done by studying the (unique) locally compact Hausdorff space $Y$ associated to $H$ in such a way that $H$ and $C_0(Y)$ are isometrically isomorphic. We are interested in various connectedness properties of $Y$. In particular, we present necessar…
We introduce two notions of amenability for a Banach algebra $\cal A$. Let $n\in \Bbb N$ and let $I$ be a closed two-sided ideal in $\cal A$, $\cal A$ is $n-I-$weakly amenable if the first cohomology group of $\cal A$ with coefficients in the n-th dual space $I^{(n)}$ is zero; i.e., $H^1({\cal A},I^{(n)})=\{0\}$. Further, $\cal A$ is n-ideally amenable if $\cal A$ is $n-I-$weakly amenable for every closed two-sided ideal $I$ in $\cal A$. We find some relationships of $n-I-$ weak amenability and $m-J-$ weak amenability for some different m and n or for different closed ideals $I$ and $J$ of $\c…
For a completely regular space $X$, let $C_B(X)$ be the normed algebra of all bounded continuous scalar-valued mappings on $X$ equipped with pointwise addition and multiplication and the supremum norm and let $C_0(X)$ be its subalgebra consisting of mappings vanishing at infinity. For a non-vanishing closed ideal $H$ of $C_B(X)$ we study properties of its spectrum $\mathfrak{sp}(H)$ which may be characterized as the unique locally compact (Hausdorff) space $Y$ such that $H$ and $C_0(Y)$ are isometrically isomorphic. We concentrate on compactness properties of $\mathfrak{sp}(H)$ and find necess…
We introduce two notions of amenability for a Banach algebra $\cal A$. Let $I$ be a closed two-sided ideal in $\cal A$, we say $\cal A$ is $I$-weakly amenable if the first cohomology group of $\cal A$ with coefficients in the dual space $I^*$ is zero; i.e., $H^1({\cal A},I^*)=\{0\}$, and, $\cal A$ is ideally amenable if $\cal A$ is $I$-weakly amenable for every closed two-sided ideal $I$ in $\cal A$. We relate these concepts to weak amenability of Banach algebras. We also show that ideal amenability is different from amenability and weak amenability. We study the $I$-weak amenability of a Bana…
We present a method of building operators on a Banach space $X$ that generate distinct operator ideals in the algebra $\mathscr{B}(X)$ of bounded linear operators on $X$. We show that there are exactly $2^\mathfrak{c}$ distinct small closed operator ideals on the Schlumprecht space and there is a chain of cardinality $\mathfrak{c}$ of small closed operator ideals on any Schreier space of finite order.
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…
We prove two dichotomy theorems about sequences of operators into $L_1$ given by random matrices. In the second theorem we assume that the entries of each random matrix form a sequence of independent, symmetric random variables. Then the corresponding sequence of operators either uniformly factor the identity operators on $\ell_1^k$ $(k\in\mathbb N$) or uniformly approximately factor through $\mathrm{c}_0$. The first theorem has a slightly weaker conclusion still related to factorization properties but makes no assumption on the random matrices. Indeed, it applies to operators defined on an ar…
For $α$ an ordinal, we investigate the class $\mathscr{SZ}_α$ consisting of all operators whose Szlenk index is an ordinal not exceeding $ω^α$. Our main result is that $\mathscr{SZ}_α$ is a closed, injective, surjective operator ideal for each $α$. We also study the relationship between the classes $\mathscr{SZ}_α$ and several well-known closed operator ideals.
Let $X$ be a completely regular topological space. We assign to each (set theoretic) ideal of $X$ an (algebraic) ideal of $C_B(X)$, the normed algebra of continuous bounded complex valued mappings on $X$ equipped with the supremum norm. We then prove several representation theorems for the assigned ideals of $C_B(X)$. This is done by associating a certain subspace of the Stone--Čech compactification $βX$ of $X$ to each ideal of $X$. This subspace of $βX$ has a simple representation, and in the case when the assigned ideal of $C_B(X)$ is closed, coincides with its spectrum as a $C^*$-subalgebra…
Following "An infinite dimensional Schur-Horn theorem and majorization theory", Journal of Functional Analysis 259 (2010) 3115-3162, this paper further studies majorization for infinite sequences. It extends to the infinite case classical results on "intermediate sequences" for finite sequence majorization. These and other infinite majorization properties are then linked to notions of infinite convexity and invariance properties under various classes of substochastic matrices to characterize arithmetic mean closed operator ideals and arithmetic mean at infinity closed operator ideals.
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…
Let $X$ be a first countable space which admits a non-trivial convergent sequence and let $\mathcal{I}$ be an analytic P-ideal. First, it is shown that the sets of $\mathcal{I}$-limit points of all sequences in $X$ are closed if and only if $\mathcal{I}$ is also an $F_σ$-ideal. Moreover, let $(x_n)$ be a sequence taking values in a Polish space without isolated points. It is known that the set $A$ of its statistical limit points is an $F_σ$-set, the set $B$ of its statistical cluster points is closed, and that the set $C$ of its ordinary limit points is closed, with $A\subseteq B\subseteq C$. …
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 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.
We investigate the quotient algebra $\mathfrak{A}_X^{\mathcal I}:=\mathcal I(X)/\overline{\mathcal F(X)}^{||\cdot||_{\mathcal I}}$ for Banach operator ideals $\mathcal I$ contained in the ideal of the compact operators, where $X$ is a Banach space that fails the $\mathcal I$-approximation property. The main results concern the nilpotent quotient algebras $\mathfrak A_X^{\mathcal{QN}_p}$ and $\mathfrak A_X^{\mathcal{SK}_p}$ for the quasi $p$-nuclear operators $\mathcal{QN}_p$ and the Sinha-Karn $p$-compact operators $\mathcal{SK}_p$. The results include the following: (i) if $X$ has cotype 2, t…
