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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Selfadjointness of $ \ast$-representations generated by positive linear functionals


Author: A. Inoue
Journal: Proc. Amer. Math. Soc. 93 (1985), 643-647
MSC: Primary 47D40; Secondary 46K10
MathSciNet review: 776195
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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}$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0776195-9
PII: S 0002-9939(1985)0776195-9
Article copyright: © Copyright 1985 American Mathematical Society