The number of closed ideals in $L(L_p)$

DOI10.4310/ACTA.2021.V227.N1.A2arXiv2003.11414MaRDI QIDQ6337410

William B. Johnson, Gideon Schechtman

Publication date: 25 March 2020

Abstract: We show that there are

2^{2^{a l e p h_{0}}}

different closed ideals in the Banach algebra

L (L_{p} (0, 1))

,

1 < p o t = 2 < i n f t y

. 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^{a l e p h_{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

{f r a k X}_{q}

of Rosenthal, which is isomorphic to a complemented subspace of

L_{q} (0, 1)

, admits such an operator.

Mathematics Subject Classification ID

Spaces of measurable functions ((L^p)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) (46E30) Spaces of operators; tensor products; approximation properties (46B28) Operator ideals (47L20) Algebras of operators on Banach spaces and other topological linear spaces (47L10) Ideals and subalgebras (46H10)

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