Automorphisms Of Operator Algebras Research Articles

Spectral and ergodic properties of the analytic generators

In this paper similarity situations between one-parameter groups of operators are characterized in terms of analytic generators and spectral subspaces, and ergodic properties of one-parameter groups are studied. These topics permit us to give “noncommutative” extensions of classical results from the theory of Hardy's Hp spaces. As major applications we mention the possibility of use of the analyticity notion from Arveson (Amer. J. Math.89 (1967), 578–642) for one-parameter groups of *-automorphisms of operator algebras with “good” ergodic properties (see Section 3), and a procedure to “improve” implementing groups of unitaries for one-parameter groups of *-automorphisms of operator algebras (see Section 5).

On groups of automorphisms of operator algebras

In section 3 we shall prove the following results: Let G be a separable locally compact abelian group, R a von Neumann algebra acting on a separable Hilbert space, and α a weakly continuous representation of G by inner *-automorphisms of R, say α(g) = ad Wg with Wg ∈ U(R). Then there is a weakly continuous unitary representation of G, by unitaries in R, implementing α if and only if the Wg's commute with each other. The result was motivated by the proof of (7), theorem 1. Suppose now Gis a discrete amenable group of *-automorphisms of a countably decomposable von Neumann algebra R. In section 3 we give a necessary and sufficient condition for the existence of a faithful normal G-invariant state on R. This generalizes a result of Hajian and Kakutani on invariant measures (2).

Algebraic Automorphisms of Operator Algebras

Algebraic Automorphisms of Operator Algebras

States and invariant maps of operator algebras

Recently a great deal of progress has been made in the study of groups of automorphisms of operator algebras and states invariant under the automorphisms. We are then given a C*-algebra Q!, a group G of automorphisms of GY satisfying certain requirements and want to draw conclusions about the invariant states and the fixed point algebra in LY under G. The present paper is concerned both with this problem and the converse one. In the latter case we are given a state p of a C*-algebra O? and want to find a group G of auto- morphisms of G! with respect to which p is invariant, such that the above mentioned theory is applicable. In some cases we find a natural abelian subalgebra 9 of OZ, which is the fixed points of a group G and a canonical G-invariant projection @ of U onto 9 such that p = (p ) 9) 0 @. Then we can prove theorems on integral decompo- sition of states. The main portion of this paper rests largely on the results of KovPcs and Sziics [13]. Given a von Neumann algebra 9 and a group G of automorphisms (i.e. *-automorphisms) of 9, we say .J% is G- finite if there is a family 5 of normal G-invariant states of 9 such that if A is a nonzero positive operator in 9 then for some p in F, p(A) # 0. We shall for simplicity say that

Unitary Groups and Automorphisms of Operator Algebras

Unitary Groups and Automorphisms of Operator Algebras

Derivations and automorphisms of operator algebras. II

Derivations and automorphisms of operator algebras. II

Derivations and automorphisms of operator algebras

The theorem that each derivation of aC*-algebra\(\mathfrak{A}\) extends to an inner derivation of the weak-operator closure ϕ(\(\mathfrak{A}\))− of\(\mathfrak{A}\) in each faithful representation ϕ of\(\mathfrak{A}\) is proved in sketch and used to study the automorphism group of\(\mathfrak{A}\) in its norm topology. It is proved that the connected component of the identity i in this group contains the open ball ℬ of radius 2 with centerl and that each automorphism in ℬ extends to an inner automorphism of ϕ(\(\mathfrak{A}\))−.

Automorphisms of operator algebras

Automorphisms of operator algebras