Every sum system is divisible. (English) Zbl 1208.46064
The Fock functor (or exponential functor) sends complex Hilbert spaces \(H\) to symmetric Fock spaces \(\Gamma(H)\), and contractions between them to contractions between the Fock spaces. It sends direct sums to tensor products: \(\Gamma(H_1\oplus H_2)=\Gamma(H_1)\otimes\Gamma(H_2)\). Therefore, it turns direct sum systems (a one-parameter family \((H_t)\) of Hilbert spaces with associative identifications \(H_s\oplus H_t=H_{s+t}\)) into product systems (a one-parameter family \((E_t)\) of Hilbert spaces with associative identifications \(E_s\otimes E_t=E_{s+t}\)). A product system is divisible (or type I) if it is generated by its units (families \((u_t)\) of vectors that factor as \(u_s\otimes u_t=u_{s+t}\)). It is type III if it has no units. (The remaining cases are type II.) Addits (a.k.a.additive units) and divisibility for direct sum systems are defined the same way. Every direct sum system is divisible, and a product system is divisible if and only if it arises from a direct sum system by the Fock functor.
A sum system is a generalization of a direct sum system, roughly speaking, a Hilbert-Schmidt perturbation of a direct sum system. Replacing for special morphisms the Fock functor with Shale’s theorem, also a sum system gives rise to a product system. (Really, a sum system is a family of real Hilbert spaces, and the Fock spaces are over their complexifications.) This functional analytic description of Tsirelson’s probabilistic contruction of type III product systems is due to [B.V.R.Bhat and R.Srinivasan, Infin.Dimens.Anal.Quantum Probab.Relat.Top.8, No.1, 1–31 (2005; Zbl 1069.46035)]. They also proved that product systems arising from divisible sum systems are either type I or type III.
Continuing his own work on type III systems (partly with Srinivasan), the author proves the following striking results:
Every sum system is divisible. Combined with B.V.R.Bhat and R.Srinivasan’s result, this shows that every product system arising from a sum system (and every generalized CCR flow) is either of type I or type III. A necessary and sufficient condition for such a product system to be of type I is obtained.
A sum system is a generalization of a direct sum system, roughly speaking, a Hilbert-Schmidt perturbation of a direct sum system. Replacing for special morphisms the Fock functor with Shale’s theorem, also a sum system gives rise to a product system. (Really, a sum system is a family of real Hilbert spaces, and the Fock spaces are over their complexifications.) This functional analytic description of Tsirelson’s probabilistic contruction of type III product systems is due to [B.V.R.Bhat and R.Srinivasan, Infin.Dimens.Anal.Quantum Probab.Relat.Top.8, No.1, 1–31 (2005; Zbl 1069.46035)]. They also proved that product systems arising from divisible sum systems are either type I or type III.
Continuing his own work on type III systems (partly with Srinivasan), the author proves the following striking results:
Every sum system is divisible. Combined with B.V.R.Bhat and R.Srinivasan’s result, this shows that every product system arising from a sum system (and every generalized CCR flow) is either of type I or type III. A necessary and sufficient condition for such a product system to be of type I is obtained.
Reviewer: Michael Skeide (Campobasso)
MSC:
| 46L55 | Noncommutative dynamical systems |
| 46L57 | Derivations, dissipations and positive semigroups in \(C^*\)-algebras |
| 47D03 | Groups and semigroups of linear operators |
| 60G51 | Processes with independent increments; Lévy processes |
| 81S05 | Commutation relations and statistics as related to quantum mechanics (general) |
Citations:
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