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These notes are the output of a decade of research on how the results about dilations of one-parameter CP-semigroups with the help of product systems, can be put forward to d-parameter semigroups - and beyond.
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S. Parrot, Unitary dilations for commuting contractions , Pacific J. Math. 34
1970
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A. Guichardet, Symmetric Hilbert spaces and related topics , Lect. Notes Math., no. 261, Springer, 1972
1972
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K.R. Parthasarathy and K. Schmidt, Positive definite kernels, continuous tensor products, and central limit theorems of probability theory , Lect. Notes Math., no. 272, Springer, 1972
1972
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W.L. Paschke, Inner product modules over B ∗ B^{*} –algebras , Trans. Amer. Math. Soc. 182
1973
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E.B. Davies, Quantum theory of open systems , Academic Press, 1976
1976
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S. Albeverio and R. Hoegh-Krohn, Frobenius theory for positive maps of von Neumann algebras , Commun. Math. Phys. 64
1978
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E. Christensen and D.E. Evans, Cohomology of operator algebras and quantum dynamical semigroups , J. London Math. Soc. 20
1979
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M. Takesaki, Theory of operator algebras I , Encyclopaedia of Mathematical Sciences, no. 124 (number V in the subseries Operator Algebras and Non-Commutative Geometry), Springer, 2002, 2nd printing of the First Edition 1979
1979
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A. Connes, Correspondences , His hand-written notes, unpublished, 1980
1980
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G.G. Kasparov, Hilbert C ∗ C^{*} –modules, theorems of Stinespring & Voiculescu , J. Operator Theory 4
1980
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L. Accardi and C. Cecchini, Conditional expectations in von Neumann algebras and a theorem of Takesaki , J. Funct. Anal. 45
1982
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A.E. Frazho, Models for noncommuting operators , J. Funct. Anal. 48
1982
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J.W. Bunce, Models for n n –tuples of noncommuting operators , J. Funct. Anal. 57
1984
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R.L. Hudson and K.R. Parthasarathy, Quantum Ito’s formula and stochastic evolutions , Commun. Math. Phys. 93
1984
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R. Longo, Solution of the factorial Stone-Weierstrass conjecture , Invent. Math. 76
1984
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B. Kümmerer, Markov dilations on W ∗ W^{*} –algebras , J. Funct. Anal. 63
1985
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M. Baillet, Y. Denizeau, and J.-F. Havet, Indice d’une esperance conditionnelle , Compositio Math. 66
1988
Earlier work this paper cites.
G. Popescu, Isometric dilations for infinite sequences of noncommuting operators , Trans. Amer. Math. Soc. 316
1989
Earlier work this paper cites.
D. Petz, An invitation to the algebra of canonical commutation relations , Leuven University Press, 1990
1990
Earlier work this paper cites.
L. Accardi, F. Fagnola, and J. Quaegebeur, A representation-free quantum stochastic calculus , J. Funct. Anal. 104
1992
Cited alongside, same era.
W. Arveson and A. Kishimoto, A note on extensions of semigroups of ∗ * –endomorphisms , Proc. Amer. Math. Soc. 116
1992
Cited alongside, same era.
B. Kümmerer and R. Speicher, Stochastic integration on the Cuntz algebra O ∞ O_{\infty} , J. Funct. Anal. 103
1992
Cited alongside, same era.
M. Schürmann, White noise on bialgebras , Lect. Notes Math., no. 1544, Springer, 1993
1993
Cited alongside, same era.
E.C. Lance, Hilbert C ∗ C^{*} –modules , Cambridge University Press, 1995
1995
Cited alongside, same era.
B.V.R. Bhat, An index theory for quantum dynamical semigroups , Trans. Amer. Math. Soc. 348
K.-J. Engel and R. Nagel, A short course on operator semigroups , Universitext, Springer, 2006
2006
Later among the works it cites.
P.S. Muhly, M. Skeide, and B. Solel, Representations of ℬ a ( E ) \mathscr{B}^{a}(E) , Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9
2006
Later among the works it cites.
B. Solel, Representations of product systems over semigroups and dilations of commuting CP maps , J. Funct. Anal. 235
2006
Later among the works it cites.
G. Abbaspour and M. Skeide, Generators of dynamical systems on Hilbert modules , Commun. Stoch. Anal. 1
2007
Later among the works it cites.
F. Fagnola and M. Skeide, Restrictions of CP-semigroups to maximal commutative subalgebras , Banach Center Publications 78
2007
Later among the works it cites.
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1996
Cited alongside, same era.
M. Skeide, Hilbert modules in quantum electro dynamics and quantum probability , Commun. Math. Phys. 192
1996
Cited alongside, same era.
Y.G. Lu, An interacting free Fock space and the arcsine law , Prob. Math. Statist. 17
1997
Cited alongside, same era.
N. Muraki, Noncommutative Brownian motion in monotone Fock space , Commun. Math. Phys. 183
1997
Cited alongside, same era.
M.V. Pimsner, A class of C ∗ C^{*} –algebras generalizing both Cuntz-Krieger algebras and crossed products by ℤ \mathbb{Z} , Free probability theory (D.V. Voiculescu, ed.), Fields Inst. Commun., no. 12, 1997, pp. 189–212
1997
Cited alongside, same era.
J. Hellmich, C. Köstler, and B. Kümmerer, Stationary quantum Markov processes as solutions of stochastic differential equations , Quantum probability (R. Alicki, M. Bozejko, and W.A. Majewski, eds.), Banach Center Publications, vol. 43, Polish Academy of Sciences — Institute of Mathematics, 1998, pp. 217–229
1998
Cited alongside, same era.
P.S. Muhly and B. Solel, Tensor algebras over C ∗ C^{*} –correspondences: representations, dilations, and C ∗ C^{*} –envelopes , J. Funct. Anal. 158
1998
Cited alongside, same era.
B.V.R. Bhat, F. Fagnola, and M. Skeide, Maximal commutative subalgebras invariant for CP-maps: (Counter-)examples , Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11
2008
Later among the works it cites.
B.V.R. Bhat, V. Liebscher, and M. Skeide, A problem of Powers and the product of spatial product systems , Quantum Probability and Infinite Dimensional Analysis — Proceedings of the 28th Conference (J.C. Garcia, R. Quezada, and S.B. Sontz, eds.), Quantum Probability and White Noise Analysis, no. XXIII, World Scientific, 2008, (arXiv: 0801.0042v1), pp. 93–106
2008
Later among the works it cites.
M. Popa, A combinatorial approach to monotonic independence over a C ∗ C^{*} –algebra , Pac. J. Math. 237
2008
Later among the works it cites.
O.M. Shalit, E 0 E_{0} –dilation of strongly commuting CP 0 –semigroups , J. Funct. Anal. 255
2008
Later among the works it cites.
V. Liebscher, Random sets and invariants for (type II) continuous tensor product systems of Hilbert spaces , Mem. Amer. Math. Soc., no. 930, American Mathematical Society, 2009, (arXiv: math.PR/0306365)
2009
Later among the works it cites.
O.M. Shalit and B. Solel, Subproduct systems , Documenta Math. 14
2009
Later among the works it cites.
B.V.R. Bhat and M. Mukherjee, Inclusion systems and amalgamated products of product systems , Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13
2010
Later among the works it cites.
D. Markiewicz and O.M. Shalit, Continuity of CP-semigroups in the point-strong operator topology , J. Operator Theory 64
2010
Later among the works it cites.
B. Sz.-Nagy, C. Foias, H. Bercovici, and Kérchy, Harmonic analysis of operators on Hilbert space , Springer Science & Business Media, 2010
2010
Later among the works it cites.
A. Viselter, Covariant representations of subproduct systems , Proc. Lond. Math. Soc. 102
2010
Later among the works it cites.
K.R. Davidson, C. Ramsey, and O.M. Shalit, The isomorphism problem for some universal operator algebras , Adv. Math. 228
2011
Later among the works it cites.
S.C. Power and B. Solel, Operator algebras associated with unitary commutation relations , J. Funct. Anal. 260
2011
Later among the works it cites.
O.M. Shalit and M. Skeide, Three commuting, unital, completely positive maps that have no minimal dilation , Integr. Equat. Oper. Th. 51
2011
Later among the works it cites.
A. Thom, A remark about the Connes fusion tensor product , Theory Appl. Categ. 25
2011
Later among the works it cites.
P. Bikram, K. Mukherjee, R. Srinivasan, and V.S. Sunder, Hilbert von Neumann Modules , Commun. Stoch. Anal. 6
2012
Later among the works it cites.
G. Landi and A. Pavlov, On orthogonal systems in Hilbert C ∗ C^{*} –modules , J. Operator Theory 68
2012
Later among the works it cites.
O.T. Margetts and R. Srinivasan, Invariants for E 0 E_{0} –semigroups on II 1 factors , Commun. Math. Phys. 323
2013
Later among the works it cites.
A. Dor-On, Tensor algebras and subproduct systems arising from Feller chains , unpublished, 2014
2014
Later among the works it cites.
A. Dor-On and D. Markiewicz, Operator algebras and subproduct systems arising from stochastic matrices , J. Funct. Anal. 267
2014
Later among the works it cites.
M. Skeide and K. Sumesh, CP-H-Extendable maps between Hilbert modules and CPH-semigroups , J. Math. Anal. Appl. 414
2014
Later among the works it cites.
A. Vernik, Dilations of CP-maps commuting according to a graph , Houston J. Math. 42
2016
Later among the works it cites.
S.P. Murugan and S. Sundar, E 0 P E_{0}^{P} –semigroups and product systems , Preprint, arXiv: math.OA/1706.03928, 2017
2017
Later among the works it cites.
M. Gerhold and M. Skeide, Interacting Fock spaces and subproduct systems , Infin. Dimens. Anal. Quantum Probab. Relat. Top. 23
2020
Closest in time.
E. Levy, (Positive) totally ordered noncommutative monoids – how noncommutative can they be? , Preprint, arXiv: 2006.00886, 2020
2020
Closest in time.