Strictly Cyclic Functionals, Reflexivity, and Hereditary Reflexivity of Operator Algebras
This paper is concerned with strictly cyclic functionals of operator algebras on Banach spaces. It is shown that if X is a reflexive Banach space and A is a norm-closed semisimple abelian subalgebra of B(X) with a strictly cyclic functional f∈X∗, then A is reflexive and hereditarily reflexive. Moreo...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2012/434308 |
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| author | Quanyuan Chen Xiaochun Fang |
| author_facet | Quanyuan Chen Xiaochun Fang |
| author_sort | Quanyuan Chen |
| collection | DOAJ |
| description | This paper is concerned with strictly cyclic functionals of operator algebras on Banach spaces. It is shown that if X is a reflexive Banach space and A is a norm-closed semisimple abelian subalgebra of B(X) with a strictly cyclic functional f∈X∗, then A is reflexive and hereditarily reflexive. Moreover, we construct a semisimple abelian operator algebra having a strictly cyclic functional but having no strictly cyclic vectors. The hereditary reflexivity of an algbra of this type can follow from theorems in this paper, but does not follow directly from the known theorems that, if a strictly cyclic operator algebra on Banach spaces is semisimple and abelian, then it is a hereditarily reflexive algebra. |
| format | Article |
| id | doaj-art-66691d82651140cfb1290865f129bbfa |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-66691d82651140cfb1290865f129bbfa2025-02-03T01:10:29ZengWileyAbstract and Applied Analysis1085-33751687-04092012-01-01201210.1155/2012/434308434308Strictly Cyclic Functionals, Reflexivity, and Hereditary Reflexivity of Operator AlgebrasQuanyuan Chen0Xiaochun Fang1Department of Mathematics, Tongji University, Shanghai 200092, ChinaDepartment of Mathematics, Tongji University, Shanghai 200092, ChinaThis paper is concerned with strictly cyclic functionals of operator algebras on Banach spaces. It is shown that if X is a reflexive Banach space and A is a norm-closed semisimple abelian subalgebra of B(X) with a strictly cyclic functional f∈X∗, then A is reflexive and hereditarily reflexive. Moreover, we construct a semisimple abelian operator algebra having a strictly cyclic functional but having no strictly cyclic vectors. The hereditary reflexivity of an algbra of this type can follow from theorems in this paper, but does not follow directly from the known theorems that, if a strictly cyclic operator algebra on Banach spaces is semisimple and abelian, then it is a hereditarily reflexive algebra.http://dx.doi.org/10.1155/2012/434308 |
| spellingShingle | Quanyuan Chen Xiaochun Fang Strictly Cyclic Functionals, Reflexivity, and Hereditary Reflexivity of Operator Algebras Abstract and Applied Analysis |
| title | Strictly Cyclic Functionals, Reflexivity, and Hereditary Reflexivity of Operator Algebras |
| title_full | Strictly Cyclic Functionals, Reflexivity, and Hereditary Reflexivity of Operator Algebras |
| title_fullStr | Strictly Cyclic Functionals, Reflexivity, and Hereditary Reflexivity of Operator Algebras |
| title_full_unstemmed | Strictly Cyclic Functionals, Reflexivity, and Hereditary Reflexivity of Operator Algebras |
| title_short | Strictly Cyclic Functionals, Reflexivity, and Hereditary Reflexivity of Operator Algebras |
| title_sort | strictly cyclic functionals reflexivity and hereditary reflexivity of operator algebras |
| url | http://dx.doi.org/10.1155/2012/434308 |
| work_keys_str_mv | AT quanyuanchen strictlycyclicfunctionalsreflexivityandhereditaryreflexivityofoperatoralgebras AT xiaochunfang strictlycyclicfunctionalsreflexivityandhereditaryreflexivityofoperatoralgebras |