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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Bibliographic Details
Published in:Abstract and Applied Analysis 2012-01, Vol.2012 (1), p.315-326-321
Main Authors: Chen, Quanyuan, Fang, Xiaochun
Format: Article
Language:English
Subjects:
Algebra
Banach space
Banach spaces
Construction
Functionals
Mathematical analysis
Mathematical problems
Operators
Power plants
Risk premiums
Theorems
Vectors (mathematics)
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Summary: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.
ISSN:1085-3375
1687-0409
DOI:10.1155/2012/434308