Geodesic
Dilations of Contraction Cocycles and Cocycle Perturbations of the Translation Group of the Line
Matematičeskie zametki, Tome 79 (2006) no. 1, pp. 3-18.

Voir la notice de l'article provenant de la source Math-Net.Ru

The class of contraction cocycles which can be dilated to unitary Markovian cocycles of a translation group $S$ on the straight line is introduced. The class of cocycle perturbations of $S$ by unitary Markovian cocycles $W$ with the property $W_t-I\in\mathscr S_2$ (the Hilbert–Schmidt class) is investigated. The results are applied to perturbations of Kolmogorov flows on hyperfinite factors generated by the algebra of canonical anticommutation relations. 
@article{MZM_2006_79_1_a0,
     author = {G. G. Amosov and A. D. Baranov},
     title = {Dilations of {Contraction} {Cocycles} and {Cocycle} {Perturbations} of the {Translation} {Group} of the {Line}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {3--18},
     publisher = {mathdoc},
     volume = {79},
     number = {1},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2006_79_1_a0/}
}
TY  - JOUR
AU  - G. G. Amosov
AU  - A. D. Baranov
TI  - Dilations of Contraction Cocycles and Cocycle Perturbations of the Translation Group of the Line
JO  - Matematičeskie zametki
PY  - 2006
SP  - 3
EP  - 18
VL  - 79
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2006_79_1_a0/
LA  - ru
ID  - MZM_2006_79_1_a0
ER  - 
%0 Journal Article
%A G. G. Amosov
%A A. D. Baranov
%T Dilations of Contraction Cocycles and Cocycle Perturbations of the Translation Group of the Line
%J Matematičeskie zametki
%D 2006
%P 3-18
%V 79
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2006_79_1_a0/
%G ru
%F MZM_2006_79_1_a0
G. G. Amosov; A. D. Baranov. Dilations of Contraction Cocycles and Cocycle Perturbations of the Translation Group of the Line. Matematičeskie zametki, Tome 79 (2006) no. 1, pp. 3-18. http://geodesic.mathdoc.fr/item/MZM_2006_79_1_a0/

[1] Amosov G. G., “Cocycle perturbation of quasifree algebraic $K$-flow leads to required asymptotic dynamics of associated completely positive semigroup”, Infin. Dimen. Anal. Quantum Probab. Rel. Top., 3:2 (2000), 237–246 | DOI | MR | Zbl

[2] Amosov G. G., “Stationary quantum stochastic processes from cohomological point of view”, Quantum Probability and Infinite Dimensional Analysis (Burg, 2001), Quantum Probab. White Noise Anal., 15, ed. W. Freudenberg, World Sci. Publ., River Edge, NJ, 2003, 29–40 | MR | Zbl

[3] Amosov G. G., “On Markovian cocycle perturbations in classical and quantum probability”, Intern. J. Math. Math. Sci., 2003, no. 54, 3443–3467 | DOI | MR | Zbl

[4] Amosov G. G., “O markovskikh vozmuscheniyakh gruppy unitarnykh operatorov, assotsiirovannoi so sluchainym protsessom so statsionarnymi prirascheniyami”, Teor. veroyatn. i ee prim., 49:1 (2004), 145–155 | MR | Zbl

[5] Accardi L., Frigerio A., Lewis J. T., “Quantum stochastic processes”, Publ. Res. Inst. Math. Sci., 18:1 (1982), 97–133 | DOI | MR | Zbl

[6] Parthasarathy K. R., An Introduction to Quantum Stochastic Calculus, Monographs in Mathematics, 85, Birkhäuser Verlag, Basel, 1992 | MR | Zbl

[7] Lindsay J. M., Wills S. J., “Markovian cocycles on operator algebras adapted to a Fock filtration”, J. Funct. Anal., 178 (2000), 269–300 | DOI | MR

[8] Liebscher V., “How to generate Markovian cocycles on boson Fock space”, Infin. Dimen. Anal. Quantum Probab. Rel. Top., 4 (2001), 215–219 | DOI | MR | Zbl

[9] Bhat B. V. R., “Minimal isometric dilations of operator cocycles”, Integral Equations Operator Theory, 42:2 (2002), 125–141 | DOI | MR | Zbl

[10] Amosov G. G., “On cocycle conjugacy of quasifree endomorphism semigroups on the CAR algebra”, J. Math. Sci., 105 (2001), 2496–2503 | DOI | MR | Zbl

[11] Bulinskii A. V., “Algebraicheskie $K$-sistemy i polupotoki sdvigov Pauersa”, UMN, 51:2 (1996), 145–146 | MR | Zbl

[12] Powers R. T., “An index theory for semigroups of *-endomorphisms of $\mathscr B(\mathscr H)$ and types II$_1$ factors”, Canad. J. Math., 40:1 (1988), 86–114 | MR | Zbl

[13] Arveson W., “Continuous analogues of Fock space”, Mem. Amer. Math. Soc., 80, no. 409, 1989, 1–66 | MR

[14] Bhat B. V. R., Skeide M., “Tensor product systems of Hilbert modules and dilations of completely positive semigroups”, Infin. Dimen. Anal. Quantum Probab. Rel. Top., 3 (2000), 519–575 | MR | Zbl

[15] Barreto S. D., Bhat B. V. R., Liebscher V., Skeide M., “Type I product systems of Hilbert modules”, J. Funct. Anal., 212 (2004), 121–181 | DOI | MR | Zbl

[16] Accardi L., Amosov G. G., Franz U., “Second quantized automorphisms of the renormalized square of white noise (RSWN) algebra”, Infin. Dimen. Anal. Quantum Probab. Rel. Top., 7 (2004), 183–194 | DOI | MR | Zbl

[17] Nikolskii N. K., Lektsii ob operatore sdviga, Nauka, M., 1980 | MR

[18] Sekefalvi-Nad B., Foiash Ch., Garmonicheskii analiz operatorov v gilbertovom prostranstve, Mir, M., 1970 | MR

[19] Amosov G. G., Baranov A. D., “On perturbations of the group of shifts on the line by unitary cocycles”, Proc. Amer. Math. Soc., 132 (2004), 3269–3273 | DOI | MR | Zbl

[20] Baranov A. D., “Izometricheskie vlozheniya prostranstv $K_\Theta$ v verkhnei poluploskosti”, Problemy matem. analiza, 21 (2000), 30–44 | Zbl

[21] Ahern P. R., Clark D. N., “On functions orthogonal to invariant subspaces”, Acta Math., 124 (1970), 191–204 | DOI | MR | Zbl

[22] Araki H., “On quasifree states of CAR and Bogoliubov automorphisms”, Contemp. Math., 62 (1985), 21–141

[23] Emch G. G., “Generalized $K$-flows”, Commun. Math. Phys., 49 (1976), 191–215 | DOI | MR | Zbl

[24] Evans D., “Completely positive quasifree maps on the CAR algebra”, Commun. Math. Phys., 70 (1979), 53–68 | DOI | MR | Zbl