Standard generalized vectors and ∗-automorphism groups of partial O∗-algebras

Trace representation of weights on partial O∗-algebras

“…Then, by (2.2), n∈N + t n Xξ n 2 < ∞, ∀X ∈ M [2] , which implies that T + ∈ S 1 (M [2] ) + = S 1 (M [2] ) + . Since T − ≡ T + − T 0 and T , T + ∈ S 1 (M [2] ), we have T − ∈ S 1 (M [2] ) + .…”

“…As a preparation we prove [2] is fully closed (it is so for example when L † (D) is closed) and T ∈ B(H) + with T H ⊂ D. Then the following statements hold:…”

“…Since T − ≡ T + − T 0 and T , T + ∈ S 1 (M [2] ), we have T − ∈ S 1 (M [2] ) + . This implies that every element T of S 1 (M [2] ) * ∩ S 1 (M [2] ) can be written as…”

“…Every element T of S 1 (M [2] ) * ∩ S 1 (M [2] ) (respectively, 1 S(M [2] ) * ∩ 1 S(M [2] )) can be written as [2] ) + (respectively, 1 S(M [2] ) + ) (j = 1, . .…”

“…Let M be a partial O * -algebra on D in H and denote by M [2] the set of all elements X of M such that the weak product X † 2 X is well-defined. We shall show that if M [2] contains the inverse N of a positive compact operator, then every weight ϕ on M + satisfying ϕ(N 2 N) < ∞ is a trace weighted by a certain positive trace-class operator T . In Sections 3.1 and 3.2 we give trace representation of uniformly continuous linear functionals and of regular weights.…”