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

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.