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)

 

 

Selfadjointness of the $ *$-representation generated by the sum of two positive linear functionals


Author: Atsushi Inoue
Journal: Proc. Amer. Math. Soc. 107 (1989), 665-674
MSC: Primary 46K10; Secondary 47D30
DOI: https://doi.org/10.1090/S0002-9939-1989-0982404-7
MathSciNet review: 982404
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \phi $ and $ \psi $ be positive linear functionals on a $ * $-algebra $ \mathcal{A}$. When the closed $ * $-representations $ {\pi _\phi }$ and $ {\pi _\psi }$ of $ \mathcal{A}$ generated by the GNS-construction for $ \phi $ and $ \psi $ are self-adjoint, we shall show that $ {\pi _{\phi + \psi }}$ is self-adjoint if and only if $ {\pi _{\phi + \psi }}{\left( \mathcal{A} \right)'}_w\mathcal{D}\left( {{\pi _{\phi + \psi }}} \right) \subset \mathcal{D}\left( {{\pi _{\phi + \psi }}} \right)$; and there exists a self-adjoint extension $ \rho $ of $ {\pi _{\phi + \psi }}$ suchthat $ \rho {\left( \mathcal{A} \right)'}_w = {\pi _{\phi + \psi }}{\left( \mathcal{A} \right)'}_w$ if and only if $ {\pi _{\phi + \psi }}{\left( \mathcal{A} \right)'}_w$ is a von Neumann algebra.


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


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46K10, 47D30

Retrieve articles in all journals with MSC: 46K10, 47D30


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0982404-7
Article copyright: © Copyright 1989 American Mathematical Society