Generalized CCR Flows

Abstract

We introduce a new construction of $E_0$-semigroups, called generalized CCR flows, with two kinds of descriptions: those arising from sum systems and those arising from pairs of $C_0$-semigroups. We get a new necessary and sufficient condition for them to be of type III, when the associated sum system is of finite index. Using this criterion, we construct examples of type III $E_0$-semigroups, which can not be distinguished from $E_0$-semigroups of type I by the invariants introduced by Boris Tsirelson. Finally, by considering the local von Neumann algebras, and by associating a type III factor to a given type III $E_0$-semigroup, we show that there exist uncountably many type III $E_0$-semigroups in this family, which are mutually non-cocycle conjugate.Comment: 45 page

A perturbation problem for the shift semigroup

“…[17] is rather an invariant for {G(s, t)} s<t as a system of topological vector spaces, it does not distinguish the E 0 -semigroup arising from M from the CCR flow. Yet, we will show that even such M sometimes produces an E 0 -semigroup of type III in the forthcoming paper [9]. Namely we will show that the resulting E 0 -semigroup is of type III if and only if ϕ / ∈ L 2 (0, ∞) and that there are uncountably many mutually non-cocycle conjugate E 0 -semigroups of type III arising in this way.…”

“…The above argument actually shows that ϕ ∈ L 2 (0, ∞) if and only if K t 2 HS = O(t) (t → +0). On the other hand, we show in the subsequent paper [9] that the E 0 -semigroup arising from e tA M is of type I if and only if ϕ ∈ L 2 (0, ∞). In view of this fact, it is tempting to conjecture that there exists a cocycle conjugacy invariant of E 0 -semigroups that is computable from the asymptotic behavior of K t 2 HS for small t.…”

“…While a perturbation argument shows that the E 0 -semigroups arising from this class of functions are cocycle conjugate to those coming from a subclass of off white noises, one cannot distinguish them from the CCR flow of index 1 by using Tsirelson's invariants discussed in [17] because the spectral density functions |M(iy)| 2 converge to 1 at infinity in this case. Yet, in the forthcoming paper [9], we show that this class of E 0 -semigroups contains uncountably many mutually non cocycle conjugate type III E 0 -semigroups. The invariant we adopt for differentiating these examples is the Murray-von Neumann type of the "local observable algebras" for an open subset U of the interval [0, 1], which may be an AFD type III factor if ϕ does not belong to L 2 (0, ∞) and U has a sufficiently complicated shape.…”

Local unitary cocycles of E0-semigroups

“…In particular, the classification of one-parameter automorphism groups of B(H) up to conjugacy can be reduced to the well-known multiplicity theory of Hahn-Hellinger of unbounded selfadjoint operators. In contrast, the classification theory of E 0 -semigroups up to cocycle conjugacy (which is the appropriate equivalence relation in this context) has proved to be much richer and full of surprises (see for example [3,[9][10][11]14,17,20,21]; we recommend Arveson's book [5] for an excellent introduction to the theory of E 0 -semigroups).…”

Mini-Workshop: Product Systems and Independence in Quantum Dynamics

“…We think that the question how independence can be perturbed, clearly, should be interesting also for understanding independence. Results like [IS07,Izu07] , Operations on certain non-commutative operator-valued random variables, Astérisque 232 (1995), 243-275.…”