Selfadjointness of the $*$-representation generated by the sum of two positive linear functionals
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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
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Additional Information
- © Copyright 1989 American Mathematical Society
- 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