Selfadjointness of $\ast$-representations generated by positive linear functionals
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- by A. Inoue PDF
- Proc. Amer. Math. Soc. 93 (1985), 643-647 Request permission
Abstract:
The first purpose of this paper is to prove that ${\pi _\tau }$ is selfadjoint when ${\pi _\phi }$ is selfadjoint and ${\pi _\psi }$ is bounded, where $\tau$ is the sum of positive linear functionals $\phi ,\psi$ on a $*$-algebra $\mathcal {A}$ and ${\pi _\tau },{\pi _\phi }$ and ${\pi _\psi }$ are $*$-representations generated by $\tau ,\phi$ and $\psi$, respectively. The second purpose is to prove that ${\pi _\phi }$ is standard, where $\phi$ is a positive linear functional on $\mathcal {A}$ such that there exists a net $\left \{ {{\phi _\alpha }} \right \}$ of positive linear functionals on $\mathcal {A}$ satisfying ${\phi _\alpha } \leqq \phi ,{\pi _{{\phi _\alpha }}}$ is bounded for all $\alpha$ and $\lim _{\alpha }{\phi _\alpha }\left ( x \right ) = \phi \left ( x \right )$ for each $x \in \mathcal {A}$.References
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Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 93 (1985), 643-647
- MSC: Primary 47D40; Secondary 46K10
- DOI: https://doi.org/10.1090/S0002-9939-1985-0776195-9
- MathSciNet review: 776195