Open Access
September 2021 The number of closed ideals in $L(L_p)$
William B. Johnson, Gideon Schechtman
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Acta Math. 227(1): 103-113 (September 2021). DOI: 10.4310/ACTA.2021.v227.n1.a2


We show that there are $2^{2^{\aleph_0}}$ different closed ideals in the Banach algebra $L(L_p(0,1)), 1 \lt p \neq 2 \lt \infty$. 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 $1 \lt q \lt 2$ the space $\mathfrak{X}_q$ of Rosenthal, which is isomorphic to a complemented subspace of $L_q(0,1)$, admits such an operator.


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William B. Johnson. Gideon Schechtman. "The number of closed ideals in $L(L_p)$." Acta Math. 227 (1) 103 - 113, September 2021.


Received: 26 March 2020; Accepted: 8 February 2021; Published: September 2021
First available in Project Euclid: 1 March 2023

Digital Object Identifier: 10.4310/ACTA.2021.v227.n1.a2

Keywords: $l_p$ spaces , ideals of operators

Vol.227 • No. 1 • September 2021
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