{"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":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":4.9000,"publicationDate":"2020-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"8","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/acta.2021.v227.n1.a2","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","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
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