Flows on a separable C∗-algebra

By a flow α on a C∗-algebra A we mean a homomorphism α: R→Aut(A) such that t 7 → αt(x) is continuous for each x ∈ A, where Aut(A) is the automorphism group of A. When α is a flow, we denote by δα the generator of α, which is a closed derivation in A, i.e., δα is a closed linear map defined on a dense ∗-subalgebra D(δα) of A into A such that δα(x) ∗ = δα(x∗) and δα(xy) = δα(x)y + xδα(y) for x, y ∈ D(δα). See [3, 4, 1, 23] for characterizations of generators and more. Given h ∈ Asa, δα+ad ih is again a generator. We denote by α(h) the flow generated by δα+ad ih. We call α(h) an inner perturbation of α. More generally, if u is an α-cocycle, i.e., u: R→U(A) is continuous such that usαs(ut) = us+t, s, t ∈ R, then t 7 → Adutαt is a flow, called a cocycle perturbation of α. Note that an inner perturbation is a cocycle perturbation; α(h) is obtained as Aduα, where u is the (differentiable) α-cocycle defined by dut/dt = utαt(ih). In general a cocycle perturbation of α is given as t 7 → Ad vα(h)t Ad v ∗ for some v ∈ U(A) and h ∈ Asa. A (non-degenerate) representation pi of the system (A,α) is called covariant if there is a unitary flow U on the Hilbert space Hpi such that AdUtpi = piαt, t ∈ R. In general we do not seem to know a good characterization for existence of covariant irreducible representations, but in the following discussion