{"title":"The number of closed ideals in $L(L_p)$","authors":"W. Johnson, G. Schechtman","doi":"10.4310/acta.2021.v227.n1.a2","DOIUrl":null,"url":null,"abstract":"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 term 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.","PeriodicalId":4,"journal":{"name":"ACS Applied Energy Materials","volume":null,"pages":null},"PeriodicalIF":5.4000,"publicationDate":"2020-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"8","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Energy Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/acta.2021.v227.n1.a2","RegionNum":3,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"CHEMISTRY, PHYSICAL","Score":null,"Total":0}
引用次数: 8
Abstract
We show that there are $2^{2^{\aleph_0}}$ different closed ideals in the Banach algebra $L(L_p(0,1))$, $1
期刊介绍：
ACS Applied Energy Materials is an interdisciplinary journal publishing original research covering all aspects of materials, engineering, chemistry, physics and biology relevant to energy conversion and storage. The journal is devoted to reports of new and original experimental and theoretical research of an applied nature that integrate knowledge in the areas of materials, engineering, physics, bioscience, and chemistry into important energy applications.
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