Selfadjoint algebras of unbounded operators. II

Source 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 ∗ {C^ \ast } -algebras. Also with the aid of this generalization of Arveson’s theorem it is shown that an operator π ( A ) ¯ \overline {\pi (A)} is affiliated with the commutant π ( A ) ′ \pi (\mathcal {A})’ of a selfadjoint representation π \pi of a ∗ ^\ast -algebra A \mathcal {A} , with A = A ∗ ∈ A A = {A^ \ast } \in \mathcal {A} , if and only if π \pi preserves a certain order structure associated with A and A \mathcal {A} . This result is then applied to obtain a characterization of standard representations of commutative ∗ ^\ast -algebras in terms of an order structure.