Selfadjoint algebras of unbounded operators. II

Abstract:

Unbounded selfadjoint representations of $^\ast$-algebras are studied. It is shown that a selfadjoint representation of the enveloping algebra of a Lie algebra can be exponentiated to give a strongly continuous unitary representation of the simply connected Lie group if and only if the representation preserves a certain order structure. This result follows from a generalization of a theorem of Arveson concerning the extensions of completely positive maps of ${C^ \ast }$-algebras. Also with the aid of this generalization of Arveson’s theorem it is shown that an operator $\overline {\pi (A)}$ is affiliated with the commutant $\pi (\mathcal {A})’$ of a selfadjoint representation $\pi$ of a $^\ast$-algebra $\mathcal {A}$, with $A = {A^ \ast } \in \mathcal {A}$, if and only if $\pi$ preserves a certain order structure associated with A and $\mathcal {A}$. This result is then applied to obtain a characterization of standard representations of commutative $^\ast$-algebras in terms of an order structure.