A NOTE ON ONE-PARAMETER GROUPS OF AUTOMORPHISMS

Let {αt:t∈R} and {βt:t∈R} be two commuting one-parameter groups of ∗-automorphisms of a von Neumann algebra M such that αt


INTRODUCTION
In [I], A. Van Daele, L. VanheeswiJck and one of the authors proved that if *-automorphlsms of avon Neann algebra M satlsfylng the equation: = + = t -t t + -t for all t R, then M can be decomposed nto subalgebras Mp and M(l-p) for a central proJectlon p in M such that t 8t on Mp and t on M(l-p), tR.
e proof of thls result is very technical and fairly long.It depends on veson's theo of spectral subspaces ([2], [3]) and as such it lacks a proper emphasis on the decomposition itself.
However, there are Important sltuatlons where It Is enough to consider the comtIng automorphlsm groups (see, for Instance [4], [5]).e purpose of thls note Is to provide a simple proof of thls decomposition result for comtlng automorphlsm groups without using the theory of spectral subspaces.We use simple algebralc technlques to obtain the proof.
Of course, In dolng so we lose the generality of the result but on the other hand we get a simple proof.For more details concerning the origin of thls operator equation, Its applications and related dec6osltlon results, we refer to [6], [7], [8], [9], [I0], [4] and [5].

MAIN RESULTS
The following is an important decomposition result for comnmtlng automorphisms Let a,B be commuting *-automorphlsms of a yon Neumann algebra -I -I M such that a + a + Then there exists a central projection p in M -I such that a 8 on Mp and a B on M(l-p).
In this paper, however, we use another version (Proposition 2.2) of the above result for the sake of clarity.The essential idea is that when we consider the *-automorphism aB on M, then by [12], N(aB-I) + R(aB-1) is o-weakly dense in M where N(aB-I) and R(aB-I) denote, respectlvely, the null space and the range space of the operators under consideration.
R(aB-I) c N(a-B) and the subalgebra L (say) generated by R(aB-I) is a two-sided ideal in M and there exists a largest central projection P0 in M such that L MP0 MPo c_ N(a-B) and P0 is a, B-invarlant (that is a(po) B(po) po ).In other words, a B on Mp 0 (see for instance [10], [II]).Similarly, by considering the orthogonal ideal L i (note that L i c_ N(aB-I)) we get a largest a,B-Invarlant central projection q0 in -I M such that a B on Mqo and the orthogonallty relation implies that (I p0)(l qo 0 For more details, we may refer to [10]  such that a + a B + 8 Among the projections p eM (respectively q eM) such -I that a on Mp (respectively a 8 on Mq) there exists a largest central projection PO in M (respectively qo in M) such that P0 and q0 are a, B- invarlant and (I p0)(l qo O.
We now come to our main result.
and [II].Thus we may have the following alternative version of Proposition 2.1.Proposition 2.2.Let a, 8 be commuting *-automorphlsms of avon Neumann algebra M -I -I THEOREM 2.3.Let {at:t e R} and {t:t e R} be two commuting one-parameter groups