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

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 $x \in \mathcal {A}$.