Closed ideals of operators on the Tsirelson and Schreier spaces

doi:10.1016/j.jfa.2020.108668

Journal of Functional Analysis

Volume 279, Issue 8, 1 November 2020, 108668

https://doi.org/10.1016/j.jfa.2020.108668 Get rights and content

Abstract

Let $B (X)$ 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 $n \in N$ . We show that the lattice of closed ideals of $B (X)$ has a very rich structure; in particular $B (X)$ 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 $P_{N} \in B (X)$ corresponding to each non-empty subset N of $N$ . A closed ideal of $B (X)$ is spatial if it is generated by $P_{N}$ for some N. We can now state our main conclusions as follows:

•
the family of spatial ideals lying strictly between the ideal of compact operators and $B (X)$ is non-empty and has no minimal or maximal elements;
•
for each pair $I ⫋ J$ of spatial ideals, there is a family ${Γ_{L} : L \in Δ}$ , where the index set Δ has the cardinality of the continuum, such that $Γ_{L}$ is an uncountable chain of spatial ideals, $⋃ Γ_{L}$ is a closed ideal that is not spatial, and $I ⫋ L ⫋ J and \overline{L + M} = J$ whenever $L, M \in Δ$ are distinct and $L \in Γ_{L}$ , $M \in Γ_{M}$ .

Section snippets

Introduction and statement of main results

Let X be a Banach space with an unconditional basis ${(b_{j})}_{j \in N}$ . For a subset N of $N$ , we write $P_{N}$ for the basis projection corresponding to N; that is, $P_{N} x = \sum_{j \in N} 〈 x, b_{j}^{⁎} 〉 b_{j}$ for each $x \in X$ , where $b_{j}^{⁎} \in X^{⁎}$ denotes the jth coordinate functional. By a spatial ideal of the Banach algebra $B (X)$ of bounded operators on X, we understand the closed, two-sided ideal generated by the basis projection $P_{N}$ for some non-empty subset N of $N$ . A spatial ideal $I$ is non-trivial if $K (X) ⫋ I ⫋ B (X),$ where $K (X)$ denotes the ideal of

Preliminaries, including the framework of the proof of Theorem 1.1

For a set N, $P (N)$ denotes its power set, while $[N]$ and ${[N]}^{< \infty}$ are the sets of infinite and finite subsets of N, respectively. We write $| N |$ for the cardinality of N; the letter $c$ denotes the cardinality of the continuum.

For two non-empty subsets M and N of $N$ , we use the notation $M < N$ to indicate that M is finite and $\max M < \min N$ . By an interval in a subset N of $R$ , we understand a set of form $J \cap N$ , where J is an interval of $R$ in the usual sense. (Note that the interval J may be open, closed or

Tsirelson's space

Following Figiel and Johnson [11], we use the term Tsirelson's space for the dual of the reflexive Banach space that Tsirelson [40] originally constructed with the property that it does not contain any of the classical sequence spaces $c_{0}$ and $ℓ_{p}$ for $1 ⩽ p < \infty$ , and we denote it by T. This convention makes no difference from the point of view of ideal lattices because T is reflexive, so the adjoint map $S \mapsto S^{⁎}, B (T) \to B (T^{⁎})$ , is an isometric, linear bijection which is anti-multiplicative in the sense that $(R S$

The Schreier spaces of finite order

The aim of this section is to establish Lemma 2.8 and Theorem 1.2(ii) for the Schreier space $X [S_{n}]$ of order $n \in N$ associated with the Schreier family $S_{n}$ , originally defined by Alspach and Argyros [1]. Their precise definition is as follows.

Definition 4.1

Let $S_{0} = {{k} : k \in N} \cup {\emptyset},$ and for $n \in N_{0}$ , recursively define $S_{n + 1} = {⋃_{i = 1}^{k} E_{i} : k \in N, E_{1}, \dots, E_{k} \in S_{n} ∖ {\emptyset}, k ⩽ \min E_{1}, E_{1} < E_{2} < \dots < E_{k}} \cup {\emptyset} .$ For $n \in N_{0}$ , the Schreier space of order n is the completion of $c_{00}$ with respect to the norm $‖ x ‖ = \sup {\sum_{j \in E} | α_{j} | : E \in S_{n} ∖ {\emptyset}} (x = {(α_{j})}_{j \in N} \in c_{00}) .$ We denote this

Some open questions

Theorem 1.2(ii) and its proof raise some natural questions. To state them concisely, let $X = X [S_{n}]$ for some $n \in N$ , and denote the closure of the ideal of operators on X which factor through $c_{0}$ by ${\overline{G}}_{c_{0}} (X)$ ; in the notation of the proof of Theorem 1.2(ii), ${\overline{G}}_{c_{0}} (X) = \overline{〈 Q 〉}$ , and the argument given in its last paragraph shows that ${\overline{G}}_{c_{0}} (X) \subseteq ⋂ {I : I is a non-trivial spatial ideal of B (X)} .$ However, we do not know whether this inclusion is proper. We also do not know whether $S (X) \subseteq {\overline{G}}_{c_{0}} (X)$ .

Another, somewhat less

Acknowledgements

The research presented in this paper was initiated when the third- and second-named authors visited Washington & Lee University, VA, in October 2015 and 2016, respectively, supported by Washington & Lee Summer Lenfest grants. It was continued when the first-named author visited the UK in February/March 2017, supported by a Scheme 2 grant (reference number 21608) from the London Mathematical Society. Kania's work has also received funding from GAČR project 19-07129Y; RVO 67985840 (Czech

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Closed ideals of operators on the Tsirelson and Schreier spaces

Abstract

Section snippets

Introduction and statement of main results

Preliminaries, including the framework of the proof of Theorem 1.1

Tsirelson's space

The Schreier spaces of finite order

Some open questions

Acknowledgements

J. Funct. Anal.

J. Funct. Anal.

J. Funct. Anal.

J. Math. Anal. Appl.

Complexity of weakly null sequences

Diss. Math.

Strictly singular operators in Tsirelson like spaces

Ill. J. Math.

A hereditarily indecomposable L∞-space that solves the scalar-plus-compact problem

Acta Math.

Two-sided ideals and congruences in the ring of bounded operators in Hilbert space

Ann. Math.

On Tsirelson's space

Isr. J. Math.

Tsirelson's Space

Closed ideals in the Banach algebra of operators on classical non-separable spaces

Math. Proc. Camb. Philos. Soc.

An example of infinite dimensional reflexive Banach space non-isomorphic to its Cartesian square

Stud. Math.

A uniformly convex Banach space which contains no ℓp

Compos. Math.

Closed ideals of operators between the classical sequence spaces

Bull. Lond. Math. Soc.

A dichotomy theorem for subsets of the power set of the natural numbers

Proc. Am. Math. Soc.

On the complemented subspaces of the Schreier spaces

Stud. Math.

Normally solvable operators and ideals associated with them

Transl. Am. Math. Soc.

Bul. Akad. Štiince RSS Mold.

Perturbation classes for semi-Fredholm operators on subprojective and superprojective spaces

Ann. Acad. Sci. Fenn., Math.

Eine Idealstruktur Banachscher Operatoralgebren

J. Reine Angew. Math.

Ideals in L(L1)

Math. Ann.

The number of closed ideals in L(Lp)

A hereditarily indecomposable $L_{\infty}$ -space that solves the scalar-plus-compact problem

A uniformly convex Banach space which contains no $ℓ_{p}$

Ideals in $L (L_{1})$

The number of closed ideals in $L (L_{p})$