A *-algebra \mathfrak{A} of linear operators with a common invariant dense domain \mathscr{D} in a Hilbert space is studied relative to the order structure given by the cone \mathfrak{A}⁺ of positive elements of \mathfrak{A} (in the sense of positive sesquilinear form on \mathscr{D}) and the ρ-topology defined as an inductive limit of the order norm ρ_A (of the subspace \mathfrak{A}_A with A as its order unit) with A∈\mathfrak{A}⁺. In particular, for those \mathfrak{A} with a countable cofinal sequence A_i in \mathfrak{A}⁺ such that A_i⁻¹∈\mathfrak{A}, the ρ-topology is proved to be order convex, any positive elements in the predual is shown to be a countable sum of vector states, and the bicommutant within the set B(\mathscr{D}, \mathscr{D}) of continuous sesquilinear forms on \mathscr{D} is shown to be the ultraweak closure of \mathfrak{A}. The structure of the commutant and the bicommutant are explicitly given in terms of their bounded operator elements which are von Neumann algebras and the commutant of each other.

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A linear operator σ on a C^*-algebra A induces a contraction \hat{σ}_φ on the Hilbert space \mathscr{H}_φ associated with a σ-invariant state φ provided σ satisfies the Schwarz inequality: σ(a^*a)≥σ(a)^*σ(a). If φ is invariant under a class \mathscr{S} of such operators, the following four properties are closely connected:
(i) abelianness of the reduction of π_φ(A) to the \hat{\mathscr{S}}_φ-invariant part of \mathscr{H}_φ,
(ii) asymptotic abelianness of φ,
(iii) abelianness of π_φ(A)'∩\hat{\mathscr{S}}_φ',
(iv) uniqueness of decompositions of φ into extremal \mathscr{S}-invariant states.
If \mathscr{S} consists of 2-positive operators, almost all the same relationships between these properties hold as for the case of automorphism groups which has already been thoroughly investigated.

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Let \mathfrak{A} be a C^*-algebra with identity, α a continuous action of a compact abelian group G as *-automorphisms of \mathfrak{A}, \mathfrak{A}^α(γ) the spectral subspace of α corresponding to γ in the dual \hat{G} of G and \mathfrak{A}^α(=\mathfrak{A}^α(0)) the fixed point algebra of α. Let δ be a closed symmetric derivation of \mathfrak{A} which commutes with α and generates a one-parameter group of *-automorphisms of \mathfrak{A}^α. We assume that the linear span of \mathfrak{A}^α(γ)^*\mathfrak{A}^α(γ) is dense in \mathfrak{A}^α for each γ∈\hat{G} and then deduce that δ is a generator on \mathfrak{A}. Some relevant material on covariant representations is also given.

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