Available in electronic format
Available in print format		 Transacrions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

An Index Theory For Quantum Dynamical Semigroups

Author(s): B. V. Rajarama Bhat
Journal: Trans. Amer. Math. Soc. 348 (1996), 561-583.
MSC (1991): Primary 46L57, 81S25, 46L55
Retrieve article in: PDF DVI PostScript
This article is available free of charge

Abstract | References | Similar articles | Additional information

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:

AFL
Accardi, L., Frigerio, A., Lewis, J.T. : Quantum Stochastic Processes, Publ. Res. Inst. Math. Sci., 18 (1982), 97-133. MR 84m:82031

AL
Alicki, R., Lendi, K. : Quantum Dynamical Semigroups and Applications, Springer Lecture Notes in Phys. 286 (1987), Berlin. MR 89b:81002

Ar1
Arveson, W. : Continuous Analogues of Fock Space, Mem. Amer. Math. Soc.,Vol. 80, No.409 (1989). MR f:47061

Ar2
------: An addition formula for the index of semigroups of endomorphisms of ${\mathcal B} (H)$, Pacific J. Math., 137 (1989), 19-36. MR 90c:47074

Ar3
------: The spectral $C^*$-algebra of an $E_0$-semigroup, Proc. of Symposia on Pure Math., 51 (1990), 1-15. MR 92c:46074

Ar4
------: Continuous analogues of Fock space II : the spectral $C^*$-algebra, J. Funct. Anal., 90 (1990), 138-205. MR 91d:46073

Ar5
------: Continuous analogues of Fock space III : singular states, J. Operator Theory, 22 (1989), 165-205. MR 90m:46118

Ar6
------: Continuous analogues of Fock space IV : essential states, Acta Math., 164 (1990), 265-300. MR 91d:46074

Ar7
------: $C^*$-algebras associated with sets of semigroups of isometries, International J. of Math. 2 (1991), 235-255. MR 92i:46082

AW
Araki, H., Woods, E.J.: Complete Boolean algebras of type I factors, Publ. Res. Inst. Math. Sci., Series A, Vol II (1966) 157-242. MR 35:748

BF
Bhat, B.V.R., Fagnola F. : On minimality of Evans-Hudson flows, (preprint).

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).

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).

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

Da1
Davies, E.B. : Quantum Theory of Open Systems, Academic Press (1976) New York. MR 58:8853

Da2
Davies, E.B. : Quantum dynamical semigroups and the neutron diffusion equation, Rep. Math. Phys. 11 (1977), 169-189. MR 58:20111

Da3
Davies, E.B. : One-Parameter Semigroups, Academic Press (1980), New York. MR 82i:47060

Da4
Davies, E.B. : Generators of dynamical semigroups, J. Funct. Anal. 34 (1979), 421-432. MR 81a:47039

Di
Dixmier, J. : Les $C^*$-algèbres et leurs représentations, Gauthier-Villars (1964) Paris. MR 80:1404

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

Em
Emch, G. G. : Minimal dilations of CP flows, $C^*$ algebras and applications to physics, Springer Lecture Notes in Math. 650 (1978), 156-159. MR 81a:46079

GKS
Gorini, V., Kossakowski, A., Sudarshan, E.C.G. : Completely positive dynamical semigroups of $n$-level systems, J. Math. Phys. 17 (1976), 821-825. MR 53:9998

Gu
Guichardet, A.: Symmetric Hilbert spaces and related topics, Springer Lecture Notes in Math. 261 (1972) Berlin. MR 58:12416

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

Li
Lindblad, G. : On the generators of quantum dynamical semigroups, Comm. Math. Phys. 48 (1976), 119-130. MR 54:1990

Me
Meyer, P.A. : Quantum Probability for Probabilists, Springer Lecture Notes in Math. 1538, (1993) Berlin. MR 94k:81152

Pa
Parthasarathy, K.R. : An Introduction to Quantum Stochastic Calculus, Monographs in Math., Birkhäuser Verlag (1992) Basil. MR 93g:81062

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

Po1
Powers, R.T. : An index theory for semigroups of endomorphisms of ${\mathcal B} (H)$ and type $II_1$ factors, Canad. J. Math., 40 (1988), 86-114. MR 89f:46116

Po2
Powers, R. T. : A non-spatial continuous semigroup of $*$-endomorphisms of ${\mathcal B} (H)$, Publ. Res. Inst. Math. Sci. 23 (1987), 1053-1069. MR 89f:46118

PP
Powers, R.T., Price, G. : Continuous spatial semigroups of $*$-endomorphisms of ${\mathcal B} ({\mathcal H} )$, Trans. Amer. Math. Soc., 321 (1990), 347-361. MR 90m:47058

PR
Powers, R.T., Robinson D. : An index for continuous semigroups of $*$ endomorphisms of ${\mathcal B} (H)$, J. Funct. Anal., 84 (1989), 85-96. MR 90f:46107

Pr
Price, G. : Shifts on type $II_1$ factors, Canad. J. Math., 39 (1987), 492-511. MR 88i:46080

Sa
Sauvageot, J-L. : Markov quantum semigroups admit covariant Markov $C^*$ dilations, Comm. Math. Phys. 106 (1986), 91-103. MR 88a:46077

St
Stinespring, W.F. : Positive functions on $C^*$ algebras, Proc. Amer. Math Soc. 6 (1955), 211-216. MR 16:1033b

SzF
Sz.-Nagy, B., Foias, C. : Harmonic Analysis of Operators on Hilbert Space, North-Holland (1970) Amsterdam. MR 37:778

Va
Varadarajan, V.S. : Geometry of Quantum Theory, Second Edition, Springer (1985) Berlin. MR 87a:81009

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

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 46L57, 81S25, 46L55

Retrieve articles in all Journals with MSC (1991): 46L57, 81S25, 46L55

Additional Information:

B. V. Rajarama Bhat
Affiliation: The Fields Institute, 222 College Street, Toronto, Ontario, Canada
Email: bhat@fields.utoronto.ca E-mail address: bhat@math.toronto.edu

DOI: 10.1090/S0002-9947-96-01520-6
PII: S 0002-9947(96)01520-6
Keywords: Completely positive maps, semigroups, Markov dilations, continuous tensor products
Received by editor(s): May 27, 1994
Additional Notes: This research was supported by a fellowship from INDAM (ITALY)
Copyright of article: Copyright 1996, American Mathematical Society

Forward Citation(s):

Information for authors on submitting citations

The following works have cited this article

Arveson, William, Minimal $E\sb 0$-semigroups., Operator algebras and their applications, Fields Inst. Commun., vol. 13, Amer. Math. Soc., Providence, RI, 1997, pp. 1-12. MR 98b:46087

Amosov, G.G.; Bulinski\u\i, A. V. , The Powers-Arveson index for quasifree dynamical semigroups. , Math. Notes 62 (1997), 781-784. MR 99g:46091

Arveson, William, The index of a quantum dynamical semigroup, J. Funct. Anal. 146 (1997), 557-588. MR 98h:46079

Arveson, William, Subalgebras of $C\sp *$-algebras. III. Multivariable operator theory, Acta Math. 181 (1998), 159-228. MR 2000e:47013

Arveson, William, Subalgebras of $C\sp *$-algebras. III. Multivariable operator theory, Acta Math. 181 (1998), 159-228. MR 2000e:47013

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google