A result on two one-parameter groups of automorphisms.

On certain pairs of automorphisms of C*-algebras

“…It was shown in [7] that if A is a von Neumann algebra and a and 0 are weakly continuous (that is, at{a) -> a, /?t(a) -* a ultraweakly as t -• 0, for each a in A), then there is a central projection p in A, independent of £ and invariant under a and /?, such that fi t \Ap = a t \Ap, /? t |4(l -p) = ot-t \A(l -p) for all t. Now suppose that A is a C*-algebra, and a and /?…”

“…It is not possible to apply the result for von Neumann algebras to the extensions of a and (3 to actions on A**, since these extended actions may not be weakly continuous. Nevertheless, it is possible to carry through most of the steps of the proof in [7] (with appropriate interpretations of spectral subspace). Proposition 2.8 of [7] is no longer valid, but a new argument may be given to show that the norm-closure of the ideal K$ denned on page 270 is an ideal.…”

“…If (7.1) holds, then e and e belong to £?i, and terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700030664 [7] On certain pairs of automorphisms of C*-algebras 203 If (7.2) holds, then e a n d e belong t o 1%, a n d …”

“…On the other hand, it was shown in [7] that if a and /? are (weakly continuous) actions of R on a von Neumann algebra A and (2) a t + a-t =Pt+ P-t for all t, then there is a central projection p, invariant under a and /?, such that (3 t \Ap = a t \Ap and /3 t \A(l -p) = c*_ t |.A(l -p) for all t. This fact (in a case where it is already clear that a and /?…”

“…are (strongly continuous) actions of R on a C*-algebra A satisfying (2), it is not immediately possible to obtain a description of a and 0 by passing to A**, because the extended actions may not be continuous. However, the proof in [7] (which uses spectral subspaces) can be modified to show that there are orthogonal ideals Ji and 1% in A, invariant under a and /?, such that…”