(question from math.stackexchange)

The theorem (Operator algebras and Quantum statistical mechanics vol. 1, Bratteli, Robinson, Thm. 3.2.46 p.261) roughly says that if one has a one parameter automorphism group $t \rightarrow\alpha_t$ acting on a von Neumann algebra $\mathfrak{M}\subset \mathcal{B}(\mathcal{H})$ which is implemented by unitary operators, i.e. $$\alpha_t (A) \overset{!}{=} U_t \cdot A \cdot U^{\dagger}_t ,\quad U_t \in \mathcal{B}(\mathcal{H})$$ such that this and that... (continuity conditions and positivity of the spectrum of the generator of $U_t$) then one can even choose $U_t$ to be in $\mathfrak{M}$.

Actually it says some more things but my question is fairly general: does the theorem only holds for one-parameter groups?

  • The things is that in the previous pages, the authors stick to generality, e.g. p.250, 251, they consider locally compact abelian groups, SNAG theorem, various definitions of spectrum, and then they switch to the group $\mathbb{R}$. It must be that it does not hold for an arbitrary locally compact abelian group but what's the fundamental difference with $\mathbb{R}$.

  • I kind of suspect there may be difficulties in defining the product of the generators of a "2 parameters" automorphisms group, but ok, that's a guess

  • The "positivity" condition is probably specific to $\mathbb{R}$...

  • 1
    $\begingroup$ The theorem holds more generally: cf. Brattelli, Robinson, first paragraph p.309, referring to the article "Energy and momentum as observables in quantum field theory" by Hans-Jürgen Borchers $\endgroup$
    – Noix07
    Apr 12, 2015 at 17:10


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