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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Extensions of $ *$-representations


Author: Andreas Kasparek
Journal: Proc. Amer. Math. Soc. 109 (1990), 1069-1077
MSC: Primary 46K10; Secondary 47D40
DOI: https://doi.org/10.1090/S0002-9939-1990-1012932-8
MathSciNet review: 1012932
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Abstract: Let $ \pi $ be a $ *$-representation of a $ *$-algebra $ \mathfrak{A}$. In general the strong commutant $ \pi {\left( \mathfrak{A} \right)' }_s$ and Theory weak commutant $ \pi {\left( \mathfrak{A} \right) }_w$ of the $ {\mathcal{O}^*}$-algebra $ \pi \left( \mathfrak{A} \right)$ do not coincide. We are looking for some methods to get extensions of $ \pi $ such that the related commutants coincide or which are even selfadjoint. In §§2 and 3 we consider so-called generated extensions that are a modification of induced extensions investigated by Borchers, Yngvason [1] and Schmüdgen [7]. In §4 let $ \mathfrak{A}$ be a $ *$-algebra and $ \mathfrak{B}$ a subset of its hermitian part $ {\mathfrak{A}_h}$ such that $ \mathfrak{A}$ is generated by $ \mathfrak{B} \cup \left\{ 1 \right\}$ as an algebra. We present a method to extend $ *$-representations $ \pi $ of such algebras, which is closely related with the extension of the symmetric operators $ \pi \left( b \right),b \in \mathfrak{B}$. In §5 we give an example that shows that the method of generated extensions is also suitable to get extensions such that the commutants of the related $ {\mathcal{O}^*}$-algebras coincide.


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DOI: https://doi.org/10.1090/S0002-9939-1990-1012932-8
Keywords: Extensions of $ *$-representations, commutants of $ {\mathcal{O}^*}$-algebras
Article copyright: © Copyright 1990 American Mathematical Society

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