Selfadjoint algebras of unbounded operators. II
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- by Robert T. Powers PDF
- Trans. Amer. Math. Soc. 187 (1974), 261-293 Request permission
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.References
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Additional Information
- © Copyright 1974 American Mathematical Society
- 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