An index theory for quantum dynamical semigroups

Transactions of the American Mathematical Society

Volumn 348, Issue 2, 1996, Pages 561-583

  Bhat, B V Rajarama ^a  

^a FIELDS INSTITUTE   (Canada)

Author keywords

Completely positive maps; Continuous tensor products; Markov dilations; Semigroups

Indexed keywords

EID: 21344440400     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0002-9947-96-01520-6     Document Type: Article

References (38)

1. [AFL] Accardi, L., Frigerio, A., Lewis, J.T. : Quantum Stochastic Processes, Publ. Res. Inst. Math. Sci., 18 (1982), 97-133. MR 84m:82031
2. [AL] Alicki, R., Lendi, K. : Quantum Dynamical Semigroups and Applications, Springer Lecture Notes in Phys. 286 (1987), Berlin. MR 89b:81002
3. [Ar1] Arveson, W. : Continuous Analogues of Fock Space, Mem. Amer. Math. Soc.,Vol. 80, No.409 (1989). MR f:47061
4. [Ar2] _: An addition formula for the index of semigroups of endomorphisms of B(H), Pacific J. Math., 137 (1989), 19-36. MR 90c:47074
5. 0-semigroup, Proc. of Symposia on Pure Math., 51 (1990), 1-15. MR 92c:46074
6. [Ar4] _: Continuous analogues of Fock space II : the spectral C*-algebra, J. Funct. Anal., 00 (1990), 138-205. MR 91d:46073
7. [Ar5] _: Continuous analogues of Fock space III : singular states, J. Operator Theory, 22 (1989), 165-205. MR 90m:46118
8. [ArG] _: Continuous analogues of Fock space IV : essential states, Acta Math., 104 (1990), 265-300. MR 91d:46074
9. [Ar7] _: C*-algebras associated with sets of semigroups of isometries, International J. of Math. 2 (1991), 235-255. MR 92i:46082
10. [AW] Araki, H., Woods, E.J.: Complete Boolean algebras of type I factors, Publ. Res. Inst. Math. Sci., Series A, Vol II (1960) 157-242. MR 35:748
11. [BF] Bhat, B.V.R., Fagnola F. : On minimality of Evans-Hudson flows, (preprint).
12. [Bh] Bhat, B.V.R. : Markov Dilations of Nonconservative Quantum Dynamical Semigroups and a Quantum Boundary Theory, Ph. D. Thesis, Indian Statistical Institute, New Delhi (1993).
13. [BP2] Bhat, B.V. R., Parthasarathy, K.R. : Markov dilations of nonconservative dynamical semigroups and a quantum boundary theory, Ann. Inst. H. Poincaré Probab. Statist., (to appear).
14. [BP3] Bhat, B.V. R., Parthsarathy, K.R. : Kolmogorov's existence theorem for Markov processes in C* algebras, Proc. Ind. Acad. Sci. Math. Sci., 103 (1994), 253-262. CMP 94:13
15. [Da1] Davies, E.B. : Quantum Theory of Open Systems, Academic Press (1976) New York. MR 58:8853
16. [Da2] Davies, E.B. : Quantum dynamical semigroups and the neutron diffusion equation, Rep. Math. Phys. 11 (1977), 169-189. MR 58:20111
17. [Da3] Davies, E.B. : One-Parameter Semigroups, Academic Press (1980), New York. MR 82i:47060
18. [Da4] Davies, E.B. : Generators of dynamical semigroups, J. Funct. Anal. 34 (1979), 421-432. MR 81a:47039
19. [Di] Dixmier, J. : Les C*-algèbres et leurs représentations, Gauthier-Villars (1964) Paris. MR 80:1404
20. [EL] Evans, D.E., Lewis, J.T. : Dilations of Irreversible Evolutions in Algebraic Quantum Theory, Comm. Dublin Inst. Adv. Stud., Ser. A no.24 (1977). MR 58:8915
21. [Em] Emch, G. G. : Minimal dilations of CP flows, C* algebras and applications to physics, Springer Lecture Notes in Math. G50 (1978), 156-159. MR 81a:46079
22. [GKS] Gorini, V., Kossakowski, A., Sudarshan, E.G.G. : Completely positive dynamical semigroups of n-level systems, J. Math. Phys. 17 (1976), 821-825. MR 53:9998
23. [Gu] Guichardet, A.: Symmetric Hilbert spaces and related topics, Springer Lecture Notes in Math. 261 (1972) Berlin. MR 58:12416
24. [HP] Hudson, R.L. Parthasarathy, K.R. : Stochastic dilations of uniformly continuous completely positive semigroups, Acta Appl. Mathematicae 2 (1984), 353-398. MR 85i:46092
25. [Li] Lindblad, G. : On the generators of quantum dynamical semigroups, Comm. Math. Phys. 48 (1976), 119-130. MR 54:1990
26. [Me] Meyer, P.A. : Quantum Probability for Probabilists, Springer Lecture Notes in Math. 1538, (1993) Berlin. MR 94k:81152
27. [Pa] Parthasarathy, K.R. : An Introduction to Quantum Stochastic Calculus, Monographs in Math., Birkhäuser Verlag (1992) Basil. MR 93g:81062
28. [PaS] Parthasarathy, K.R., Schmidt, K. : Positive Definite Kernels, Continuous Tensor Products and Central Limit Theorems of Probability Theory, Springer Lecture Notes in Math. 272 (1972) Berlin. MR 58:29849
29. 1 factors, Canad. J. Math., 40 (1988), 86-114. MR 89f:46116
30. [Po2] Powers, R. T. : A non-spatial continuous semigroup of *-endomorphisins of B(H), Publ. Res. Inst. Math. Sci. 23 (1987), 1053-1069. MR 89f:46118
31. [PP] Powers, R.T., Price, G. : Continuous spatial semigroups of *-endomorphisms of B(H), Trans. Amer. Math. Soc., 321 (1990), 347-361. MR 90m:47058
32. [PR] Powers, R.T., Robinson D. : An index for continuous semigroups of * endomorphisms of B(H), J. Funct. Anal., 84 (1989), 85-96. MR 90f:46107
33. 1 factors, Canad. J. Math., 39 (1987), 492-511. MR 88i:46080
34. [Sa] Sauvageot, J-L. : Markov quantum semigroups admit covariant Markov C* dilations, Comm. Math. Phys. 106 (1986), 91-103. MR 88a:46077
35. [St] Stinespring, W.F. : Positive functions on C* algebras, Proc. Amer. Math Soc. 6 (1955), 211-216. MR 16:1033b
36. [SzF] Sz.-Nagy, B., Foias, C. : Harmonic Analysis of Operators on Hilbert Space, North-Holland (1970) Amsterdam. MR 37:778
37. [Va] Varadarajan, V.S. : Geometry of Quantum Theory, Second Edition, Springer (1985) Berlin. MR 87a:81009
38. [ViS] Vincent-Smith, G. F. : Dilations of a dissipative quantum dynamical system to a quantum Markov process, Proc. London Math. Soc.(3), 49 (1984), 58-72. MR 85i:46086