Extensions of $*$-representations

Abstract:

Let $\pi$ be a $*$-representation of a $*$-algebra $\mathfrak {A}$. In general the strong commutant $\pi {\left ( \mathfrak {A} \right )’ }_s$ and Theory weak commutant $\pi {\left ( \mathfrak {A} \right ) }_w$ of the ${\mathcal {O}^*}$-algebra $\pi \left ( \mathfrak {A} \right )$ do not coincide. We are looking for some methods to get extensions of $\pi$ such that the related commutants coincide or which are even selfadjoint. In §§2 and 3 we consider so-called generated extensions that are a modification of induced extensions investigated by Borchers, Yngvason [1] and Schmüdgen [7]. In §4 let $\mathfrak {A}$ be a $*$-algebra and $\mathfrak {B}$ a subset of its hermitian part ${\mathfrak {A}_h}$ such that $\mathfrak {A}$ is generated by $\mathfrak {B} \cup \left \{ 1 \right \}$ as an algebra. We present a method to extend $*$-representations $\pi$ of such algebras, which is closely related with the extension of the symmetric operators $\pi \left ( b \right ),b \in \mathfrak {B}$. In §5 we give an example that shows that the method of generated extensions is also suitable to get extensions such that the commutants of the related ${\mathcal {O}^*}$-algebras coincide.