A reflexive Banach space whose algebra of operators is not a Grothendieck space
By a result of Johnson, the Banach space F=(⨁n=1∞ℓ1n)ℓ∞ contains a complemented copy of ℓ1. We identify F with a complemented subspace of the space of (bounded, linear) operators on the reflexive space (⨁n=1∞ℓ1n)ℓp (p∈(1,∞)), thus solving negatively the problem posed in the monograph of Diestel and...
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| Published in: | Journal of mathematical analysis and applications Vol. 401; no. 1; pp. 242 - 243 |
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| Abstract | By a result of Johnson, the Banach space F=(⨁n=1∞ℓ1n)ℓ∞ contains a complemented copy of ℓ1. We identify F with a complemented subspace of the space of (bounded, linear) operators on the reflexive space (⨁n=1∞ℓ1n)ℓp (p∈(1,∞)), thus solving negatively the problem posed in the monograph of Diestel and Uhl which asks whether the space of operators on a reflexive Banach space is Grothendieck. |
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| AbstractList | By a result of Johnson, the Banach space F=(⨁n=1∞ℓ1n)ℓ∞ contains a complemented copy of ℓ1. We identify F with a complemented subspace of the space of (bounded, linear) operators on the reflexive space (⨁n=1∞ℓ1n)ℓp (p∈(1,∞)), thus solving negatively the problem posed in the monograph of Diestel and Uhl which asks whether the space of operators on a reflexive Banach space is Grothendieck. |
| Author | Kania, Tomasz |
| Author_xml | – sequence: 1 givenname: Tomasz surname: Kania fullname: Kania, Tomasz email: t.kania@lancaster.ac.uk organization: Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster LA1 4YF, United Kingdom |
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| Cites_doi | 10.2140/pjm.1970.32.551 10.1007/BF02762804 10.4064/sm-75-2-193-216 10.4153/CJM-1953-017-4 10.1007/BF01459739 |
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| Keywords | Space of bounded operators Grothendieck space Banach space Reflexive space |
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| References | Diestel (br000010) 1984; vol. 92 Diestel, Uhl Jr (br000015) 1977; vol. 15 Tong (br000040) 1970; 32 Grothendieck (br000020) 1953; 5 Soybaş (br000035) 2010; 4 Bourgain (br000005) 1983; 75 Pfitzner (br000030) 1994; 298 Johnson (br000025) 1972; 13 Diestel (10.1016/j.jmaa.2012.12.017_br000015) 1977; vol. 15 Tong (10.1016/j.jmaa.2012.12.017_br000040) 1970; 32 Diestel (10.1016/j.jmaa.2012.12.017_br000010) 1984; vol. 92 Bourgain (10.1016/j.jmaa.2012.12.017_br000005) 1983; 75 Grothendieck (10.1016/j.jmaa.2012.12.017_br000020) 1953; 5 Pfitzner (10.1016/j.jmaa.2012.12.017_br000030) 1994; 298 Johnson (10.1016/j.jmaa.2012.12.017_br000025) 1972; 13 Soybaş (10.1016/j.jmaa.2012.12.017_br000035) 2010; 4 |
| References_xml | – volume: 75 start-page: 193 year: 1983 end-page: 216 ident: br000005 article-title: is a Grothendieck space publication-title: Studia Math. contributor: fullname: Bourgain – volume: 298 start-page: 349 year: 1994 end-page: 371 ident: br000030 article-title: Weak compactness in the dual of a C publication-title: Math. Ann. contributor: fullname: Pfitzner – volume: vol. 15 year: 1977 ident: br000015 publication-title: Vector Measures contributor: fullname: Uhl Jr – volume: vol. 92 year: 1984 ident: br000010 publication-title: Sequences and Series in Banach Spaces contributor: fullname: Diestel – volume: 32 start-page: 551 year: 1970 end-page: 559 ident: br000040 article-title: Diagonal submatrices of matrix maps publication-title: Pacific J. Math. contributor: fullname: Tong – volume: 5 start-page: 129 year: 1953 end-page: 173 ident: br000020 article-title: Sur les applications linéaires faiblement compactes d’espaces du type publication-title: Canad. J. Math. contributor: fullname: Grothendieck – volume: 13 start-page: 301 year: 1972 end-page: 310 ident: br000025 article-title: A complementary universal conjugate Banach space and its relation to the approximation problem publication-title: Israel J. Math. contributor: fullname: Johnson – volume: 4 start-page: 167 year: 2010 end-page: 173 ident: br000035 article-title: Some remarks on the weak sequential completeness in the algebra of integral operators publication-title: Int. J. Math. Anal. contributor: fullname: Soybaş – volume: 32 start-page: 551 issue: 2 year: 1970 ident: 10.1016/j.jmaa.2012.12.017_br000040 article-title: Diagonal submatrices of matrix maps publication-title: Pacific J. Math. doi: 10.2140/pjm.1970.32.551 contributor: fullname: Tong – volume: 13 start-page: 301 issue: 3–4 year: 1972 ident: 10.1016/j.jmaa.2012.12.017_br000025 article-title: A complementary universal conjugate Banach space and its relation to the approximation problem publication-title: Israel J. Math. doi: 10.1007/BF02762804 contributor: fullname: Johnson – volume: 75 start-page: 193 year: 1983 ident: 10.1016/j.jmaa.2012.12.017_br000005 article-title: H∞ is a Grothendieck space publication-title: Studia Math. doi: 10.4064/sm-75-2-193-216 contributor: fullname: Bourgain – volume: 5 start-page: 129 year: 1953 ident: 10.1016/j.jmaa.2012.12.017_br000020 article-title: Sur les applications linéaires faiblement compactes d’espaces du type C(K) publication-title: Canad. J. Math. doi: 10.4153/CJM-1953-017-4 contributor: fullname: Grothendieck – volume: 4 start-page: 167 issue: 4 year: 2010 ident: 10.1016/j.jmaa.2012.12.017_br000035 article-title: Some remarks on the weak sequential completeness in the algebra of integral operators publication-title: Int. J. Math. Anal. contributor: fullname: Soybaş – volume: 298 start-page: 349 issue: 2 year: 1994 ident: 10.1016/j.jmaa.2012.12.017_br000030 article-title: Weak compactness in the dual of a C∗-algebra is determined commutatively publication-title: Math. Ann. doi: 10.1007/BF01459739 contributor: fullname: Pfitzner – volume: vol. 15 year: 1977 ident: 10.1016/j.jmaa.2012.12.017_br000015 contributor: fullname: Diestel – volume: vol. 92 year: 1984 ident: 10.1016/j.jmaa.2012.12.017_br000010 contributor: fullname: Diestel |
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| Snippet | By a result of Johnson, the Banach space F=(⨁n=1∞ℓ1n)ℓ∞ contains a complemented copy of ℓ1. We identify F with a complemented subspace of the space of... |
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| SubjectTerms | Banach space Grothendieck space Reflexive space Space of bounded operators |
| Title | A reflexive Banach space whose algebra of operators is not a Grothendieck space |
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