Half-Sided Translations and tye Type of von Neumann Algebras

Let M be a von Neumann algebra acting on a Hilbert space H and H a cyclic and separating vector for M. If there exists a half-sided translation for M, i.e. a continuous unitary group U(t) with U(t)=, a non-negative spectrum fulfilling Ad U(t)M M for t 0 (or 0), then we will show that either M is of type III₁ or U(t) is trivial.