Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



An asymptotic double commutant theorem for $ C\sp{\ast} $-algebras

Author: Donald W. Hadwin
Journal: Trans. Amer. Math. Soc. 244 (1978), 273-297
MSC: Primary 47A99; Secondary 46L05, 47B47
MathSciNet review: 506620
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Abstract: An asymptotic version of von Neumann's double commutant theorem is proved in which $ {C^{\ast}}$-algebras play the role of von Neumann algebras. This theorem is used to investigate asymptotic versions of similarity, reflexivity, and reductivity. It is shown that every nonseparable, norm closed, commutative, strongly reductive algebra is selfadjoint. Applications are made to the study of operators that are similar to normal (subnormal) operators. In particular, if T is similar to a normal (subnormal) operator and $ \pi $ is a representation of the $ {C^{\ast}}$-algebra generated by t, then $ \pi (T)$ is similar to a normal (subnormal) operator.

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Keywords: Double commutant, reflexive operator, reductive operator, approximate double commutant, approximately similar, approximately reflexive, strongly reductive, decomposable function, approximate equivalence, normal operator, subnormal operator, representation
Article copyright: © Copyright 1978 American Mathematical Society