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Transactions of the American Mathematical Society

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

 

 

Selfadjoint algebras of unbounded operators. II


Author: Robert T. Powers
Journal: Trans. Amer. Math. Soc. 187 (1974), 261-293
MSC: Primary 46K10
DOI: https://doi.org/10.1090/S0002-9947-1974-0333743-8
MathSciNet review: 0333743
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Abstract: Unbounded selfadjoint representations of $ ^\ast $-algebras are studied. It is shown that a selfadjoint representation of the enveloping algebra of a Lie algebra can be exponentiated to give a strongly continuous unitary representation of the simply connected Lie group if and only if the representation preserves a certain order structure. This result follows from a generalization of a theorem of Arveson concerning the extensions of completely positive maps of $ {C^ \ast }$-algebras. Also with the aid of this generalization of Arveson's theorem it is shown that an operator $ \overline {\pi (A)} $ is affiliated with the commutant $ \pi (\mathcal{A})'$ of a selfadjoint representation $ \pi $ of a $ ^\ast $-algebra $ \mathcal{A}$, with $ A = {A^ \ast } \in \mathcal{A}$, if and only if $ \pi $ preserves a certain order structure associated with A and $ \mathcal{A}$. This result is then applied to obtain a characterization of standard representations of commutative $ ^\ast $-algebras in terms of an order structure.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0333743-8
Article copyright: © Copyright 1974 American Mathematical Society