An Index Theory For Quantum Dynamical Semigroups

Bhat, B.

doi:10.1090/S0002-9947-96-01520-6

An Index Theory For Quantum Dynamical Semigroups
HTML articles powered by AMS MathViewer

by B. V. Rajarama Bhat

Trans. Amer. Math. Soc. 348 (1996), 561-583

DOI: https://doi.org/10.1090/S0002-9947-96-01520-6

PDF | Request permission

Abstract:

W. Arveson showed a way of associating continuous tensor product systems of Hilbert spaces with endomorphism semigroups of type I factors. We do the same for general quantum dynamical semigroups through a dilation procedure. The product system so obtained is the index and its dimension is a numerical invariant for the original semigroup.

References

Luigi Accardi, Alberto Frigerio, and John T. Lewis, Quantum stochastic processes, Publ. Res. Inst. Math. Sci. 18 (1982), no. 1, 97–133. MR 660823, DOI 10.2977/prims/1195184017
R. Alicki and K. Lendi, Quantum dynamical semigroups and applications, Lecture Notes in Physics, vol. 286, Springer-Verlag, Berlin, 1987. MR 911678, DOI 10.1007/3-540-18276-4_{5}
Arveson, W. : Continuous Analogues of Fock Space, Mem. Amer. Math. Soc.,Vol. 80, No.409 (1989).
William Arveson, An addition formula for the index of semigroups of endomorphisms of $B(H)$, Pacific J. Math. 137 (1989), no. 1, 19–36. MR 983326, DOI 10.2140/pjm.1989.137.19
William Arveson, The spectral $C^*$-algebra of an $E_0$-semigroup, Operator theory: operator algebras and applications, Part 1 (Durham, NH, 1988) Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1990, pp. 1–15. MR 1077377, DOI 10.1090/pspum/051.1/1077377
William Arveson, Continuous analogues of Fock space. II. The spectral $C^*$-algebra, J. Funct. Anal. 90 (1990), no. 1, 138–205. MR 1047579, DOI 10.1016/0022-1236(90)90082-V
William Arveson, Continuous analogues of Fock space. III. Singular states, J. Operator Theory 22 (1989), no. 1, 165–205. MR 1026080
William Arveson, Continuous analogues of Fock space. IV. Essential states, Acta Math. 164 (1990), no. 3-4, 265–300. MR 1049159, DOI 10.1007/BF02392756
William Arveson, $C^*$-algebras associated with sets of semigroups of isometries, Internat. J. Math. 2 (1991), no. 3, 235–255. MR 1104118, DOI 10.1142/S0129167X91000156
Huzihiro Araki and E. J. Woods, Addenda: Complete Boolean algebras of type $\textrm {I}$ factors, Publ. Res. Inst. Math. Sci. Ser. A 2 (1966/1967), 451–452. MR 0209852, DOI 10.2977/prims/1195195771
Bhat, B.V.R., Fagnola F. : On minimality of Evans-Hudson flows, (preprint).
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).
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).
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.
E. B. Davies, Quantum theory of open systems, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0489429
E. B. Davies, Quantum dynamical semigroups and the neutron diffusion equation, Rep. Mathematical Phys. 11 (1977), no. 2, 169–188. MR 503331, DOI 10.1016/0034-4877(77)90059-3
Edward Brian Davies, One-parameter semigroups, London Mathematical Society Monographs, vol. 15, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980. MR 591851
E. B. Davies, Generators of dynamical semigroups, J. Functional Analysis 34 (1979), no. 3, 421–432. MR 556264, DOI 10.1016/0022-1236(79)90085-5
Dunham Jackson, A class of orthogonal functions on plane curves, Ann. of Math. (2) 40 (1939), 521–532. MR 80, DOI 10.2307/1968936
D. E. Evans and J. T. Lewis, Dilations of irreversible evolutions in algebraic quantum theory, Communications Dublin Inst. Advanced Studies. Ser. A 24 (1977), v+104. MR 489494
Gérard G. Emch, Minimal dilations of CP-flows, $\textrm {C}^{\ast }$-algebras and applications to physics (Proc. Second Japan-USA Sem., Los Angeles, Calif., 1977) Lecture Notes in Math., vol. 650, Springer, Berlin, 1978, pp. 156–159. MR 504761
Vittorio Gorini, Andrzej Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of $N$-level systems, J. Mathematical Phys. 17 (1976), no. 5, 821–825. MR 406206, DOI 10.1063/1.522979
Alain Guichardet, Symmetric Hilbert spaces and related topics, Lecture Notes in Mathematics, Vol. 261, Springer-Verlag, Berlin-New York, 1972. Infinitely divisible positive definite functions. Continuous products and tensor products. Gaussian and Poissonian stochastic processes. MR 0493402, DOI 10.1007/BFb0070306
R. L. Hudson and K. R. Parthasarathy, Stochastic dilations of uniformly continuous completely positive semigroups, Acta Appl. Math. 2 (1984), no. 3-4, 353–378. MR 753700, DOI 10.1007/BF02280859
G. Lindblad, On the generators of quantum dynamical semigroups, Comm. Math. Phys. 48 (1976), no. 2, 119–130. MR 413878, DOI 10.1007/BF01608499
Paul-André Meyer, Quantum probability for probabilists, Lecture Notes in Mathematics, vol. 1538, Springer-Verlag, Berlin, 1993. MR 1222649, DOI 10.1007/978-3-662-21558-6
K. R. Parthasarathy, An introduction to quantum stochastic calculus, Monographs in Mathematics, vol. 85, Birkhäuser Verlag, Basel, 1992. MR 1164866, DOI 10.1007/978-3-0348-8641-3
K. R. Parthasarathy and K. Schmidt, Positive definite kernels, continuous tensor products, and central limit theorems of probability theory, Lecture Notes in Mathematics, Vol. 272, Springer-Verlag, Berlin-New York, 1972. MR 0622034
Robert T. Powers, An index theory for semigroups of $^*$-endomorphisms of ${\scr B}({\scr H})$ and type $\textrm {II}_1$ factors, Canad. J. Math. 40 (1988), no. 1, 86–114. MR 928215, DOI 10.4153/CJM-1988-004-3
Robert T. Powers, A nonspatial continuous semigroup of $*$-endomorphisms of ${\mathfrak {B}}({\mathfrak {H}})$, Publ. Res. Inst. Math. Sci. 23 (1987), no. 6, 1053–1069. MR 935715, DOI 10.2977/prims/1195175872
Robert T. Powers and Geoffrey Price, Continuous spatial semigroups of $*$-endomorphisms of ${\mathfrak {B}}({\mathfrak {H}})$, Trans. Amer. Math. Soc. 321 (1990), no. 1, 347–361. MR 974524, DOI 10.1090/S0002-9947-1990-0974524-0
Robert T. Powers and Derek W. Robinson, An index for continuous semigroups of $*$-endomorphisms of ${\mathfrak {B}}({\mathfrak {H}})$, J. Funct. Anal. 84 (1989), no. 1, 85–96. MR 999489, DOI 10.1016/0022-1236(89)90111-0
Geoffrey L. Price, Shifts on type $\textrm {II}_1$ factors, Canad. J. Math. 39 (1987), no. 2, 492–511. MR 899846, DOI 10.4153/CJM-1987-021-2
Jean-Luc Sauvageot, Markov quantum semigroups admit covariant Markov $C^\ast$-dilations, Comm. Math. Phys. 106 (1986), no. 1, 91–103. MR 853979, DOI 10.1007/BF01210927
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Béla Sz.-Nagy and Ciprian Foiaş, Analyse harmonique des opérateurs de l’espace de Hilbert, Masson et Cie, Paris; Akadémiai Kiadó, Budapest, 1967 (French). MR 0225183
V. S. Varadarajan, Geometry of quantum theory, 2nd ed., Springer-Verlag, New York, 1985. MR 805158
G. F. Vincent-Smith, Dilation of a dissipative quantum dynamical system to a quantum Markov process, Proc. London Math. Soc. (3) 49 (1984), no. 1, 58–72. MR 743370, DOI 10.1112/plms/s3-49.1.58