Abstract
Let M be a von Neumann algebra acting on a Hilbert space H and Ω∞ 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)Ω=Ω, 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.
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Borchers, H.J. Half-Sided Translations and tye Type of von Neumann Algebras. Letters in Mathematical Physics 44, 283–290 (1998). https://doi.org/10.1023/A:1007400109519
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DOI: https://doi.org/10.1023/A:1007400109519