Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
MathSciNet review: 1012932
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] H. J. Borchers and J. Yngvason, On the algebra of field operators. The weak commutant and integral decomposition of states, Comm. Math. Phys. 42 (1975), 231-252. MR 0377550 (51:13721)
  • [2] N. K. Nikol' skii, Treatise on the shift operator, Springer-Verlag, Berlin, New York, 1986. MR 827223 (87i:47042)
  • [3] S. C. Power, Hankel operators on Hilbert space, Bull. London Math. Soc. 12 (1980), 422-442. MR 593961 (82a:47030)
  • [4] R. T. Powers, Selfadjoint algebras of unbounded operators, Comm. Math. Phys. 21 (1971), 85-124. MR 0283580 (44:811)
  • [5] K. Schmüdgen, On domains of powers of closed symmetric operators, J. Operator Theory 9 (1983), 53-75. MR 695940 (85h:47030a)
  • [6] -, Unbounded commutants and intertwining spaces of unbounded symmetric operators and $ *$-representations, J. Funct. Anal. 71 (1987), 47-68. MR 879700 (88d:47059)
  • [7] -, Unbounded operator algebras and representations, Akademie-Verlag, Berlin, 1989.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46K10, 47D40

Retrieve articles in all journals with MSC: 46K10, 47D40

Additional Information

Keywords: Extensions of $ *$-representations, commutants of $ {\mathcal{O}^*}$-algebras
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society