[Paper Review] The number of closed ideals in $L(L_p)$

William B. Johnson, Gideon Schechtman|arXiv (Cornell University)|Mar 25, 2020

Advanced Banach Space Theory11 references4 citations

TL;DR

This paper resolves a long-standing problem in operator theory by proving that the Banach algebra $ L(L_p) $, for $ 1 < p \neq 2 < \infty $, contains exactly $ 2^{2^{\aleph_0}} $ distinct closed ideals. The authors construct a new class of large, non-singular ideals using a special operator on Rosenthal's $ \mathfrak{X}_q $ space ($ 1 < q < 2 $) with asymptotic properties that generate a continuum of ideals, distinct from previously known ideals including those containing strictly singular operators.

ABSTRACT

We show that there are $2^{2^{\aleph_0}}$ different closed ideals in the Banach algebra $L(L_p(0,1))$, $1

Motivation & Objective

• To resolve A. Pietsch's 1978 open problem on the number of closed ideals in $ L(L_p) $ for $ 1 < p \neq 2 < \infty $.
• To construct new large closed ideals in $ L(L_p) $ that are not contained in the strictly singular operators and do not contain projections onto non-Hilbertian subspaces.
• To establish a general criterion for a Banach space with an unconditional basis to admit $ 2^{2^{\aleph_0}} $ closed ideals via a single operator with specific asymptotic behavior.
• To show that Rosenthal's $ \mathfrak{X}_q $ space for $ 1 < q < 2 $ supports such an operator, thereby enabling the construction of the full ideal continuum.
• To prove that $ L(L_p) $ also contains exactly $ 2^{2^{\aleph_0}} $ small (strictly singular) closed ideals, using duality and complementation.

Proposed method

• Introduce a family $ \mathcal{C} $ of $ 2^{2^{\aleph_0}} $ infinite subsets of $ \mathbb{N} $, pairwise intersecting finitely, to index the ideals.
• Define a special operator $ T $ on $ \mathfrak{X}_q $ with asymptotic properties ensuring that the operators $ T_\alpha $, indexed by $ \alpha \in \mathcal{C} $, generate distinct closed ideals.
• Use the factorization constant $ \gamma_2(T) $ and Banach–Mazur distances to control the asymptotic behavior of $ T $ on block subspaces of $ \mathfrak{X}_q $.
• Construct a complemented subspace $ Y \subset \mathfrak{X}_q $ isomorphic to $ \ell_q $, and define an operator $ U: Y \to \mathfrak{X}_q $ as the identity on block subspaces to preserve norms and structure.
• Apply duality and complementation: since $ \mathfrak{X}_q \subset L_q $, and $ L_p $ is dual to $ L_q $ for $ 1/p + 1/q = 1 $, the results extend from $ \mathfrak{X}_q $ to $ L_p $.
• Use a contradiction argument involving strictly singular operators not in the closure of certain ideals to prove that the constructed ideals do not contain the strictly singular operators.

Experimental results

Research questions

• RQ1 Does $ L(L_p) $, for $ 1 < p \neq 2 < \infty $, contain a continuum of closed ideals?
• RQ2 Can large closed ideals in $ L(L_p) $ be constructed that are not contained in the strictly singular operators and do not contain projections onto non-Hilbertian subspaces?
• RQ3 Is there a general criterion for a Banach space with an unconditional basis to admit $ 2^{2^{\aleph_0}} $ closed ideals based on the existence of a single operator with specific asymptotic properties?
• RQ4 Do the ideals constructed in $ L(L_p) $ contain the strictly singular operators?
• RQ5 How many small (strictly singular) closed ideals does $ L(L_p) $ contain?

Key findings

• The Banach algebra $ L(L_p) $, for $ 1 < p \neq 2 < \infty $, contains exactly $ 2^{2^{\aleph_0}} $ distinct closed ideals.
• The constructed ideals are large but not contained in the strictly singular operators, and do not contain projections onto subspaces isomorphic to $ \ell_p $.
• The ideals are not contained in $ \overline{\Gamma_2}(L_p) $, the closure of operators factoring through a Hilbert space, and in fact $ \overline{\Gamma_2}(L_p) $ does not contain the strictly singular operators.
• For $ 1 < q < 2 $, the space $ \mathfrak{X}_q $ admits an operator $ T $ with the required asymptotic properties to generate $ 2^{2^{\aleph_0}} $ distinct ideals in $ L(\mathfrak{X}_q) $.
• By duality and complementation, the same result holds for $ L(L_p) $, $ 1 < p \neq 2 < \infty $.
• The space $ L(L_p) $ also contains exactly $ 2^{2^{\aleph_0}} $ small closed ideals, i.e., strictly singular operators, resolving a further open question.