Abstract
We generalize Bhat's construction of product systems of Hilbert spaces from E0-semigroups on B(H) for some Hilbert space H to the construction of product systems of Hilbert modules from E0-semigroups on Ba(E) for some Hilbert module E. As a byproduct we find the representation theory for Ba(E) if E has a unit vector. We establish a necessary and sufficient criterion for the conditional expectation generated by the unit vector to define a weak dilation of a CP-semigroup in the sense of [1]. Finally, we also show that white noises on general von Neumann algebras in the sense of [2] can be extended to white noises on a Hilbert module.
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REFERENCES
B. V. R. Bhat and M. Skeide, "Tensor product systems of Hilbert modules and dilations of completely positive semigroups," Infinite-Dimensional Analysis, Quantum Probability and Related Topics, 3 (2000), 519–575.
B. Kümmerer, "Markov dilations on W ?-algebras," J. Funct. Anal., 63 (1985), 139–177.
M. Skeide, Hilbert Modules and Applications in Quantum Probability, Habilitationsschrift, Cottbus, 2001.
W. Arveson, Continuous Analogues of Fock Space, vol. 409, Mem. Amer. Math. Soc, Amer. Math. Soc., Providence (R.I.), 1989.
B. V. R. Bhat, "An index theory for quantum dynamical semigroups," Trans. Amer. Math. Soc., 348 (1996), 561–583.
G. G. Kasparov, "Hilbert C ?-modules and theorems of Steinspring and Voiculescu," J. Operator Theory, 4 (1980), 133–150.
E. C. Lance, Hilbert C ? -Modules, Cambridge University Press, Cambridge, 1995.
B. V. R. Bhat and K. R. Parthasarathy, "Kolmogorov's existence theorem for Markov processes in C ?-algebras," Proc. Indian Acad. Sci. (Math. Sci.), 104 (1994), 253–262.
B. V. R. Bhat and K. R. Parthasarathy, "Markov dilations of nonconservative dynamical semigroups and a quantum boundary theory," Ann. I.H.P. Prob. Stat., 31 (1995), 601–651.
M. Skeide, "Generalized matrix C ?-algebras and representations of Hilbert modules," Math. Proc. Royal Irish Acad., 100A (2000), 11–38.
J. Cuntz, "Simple C ?-algebras generated by isometries," Comm. Math. Phys., 57 (1977), 173–185.
S. D. Barreto, B. V. R. Bhat, V. Liebscher, and M. Skeide, Type I Product Systems of Hilbert Modules, Preprint, BTU Cottbus, Cottbus, 2000.
M. Skeide, The Index of White Noises and Their Product Systems, Preprint, Centro Vito Volterra, Rome, 2001.
D. Goswami and K. B. Sinha, "Hilbert modules and stochastic dilation of a quantum dynamical semigroup on a von Neumann algebra," Comm. Math. Phys., 205 (1999), 377–403.
M. Skeide, "Quantum stochastic calculus on full Fock modules," J. Funct. Anal., 173 (2000), 401–452.
J.-L. Sauvageot, "Markov quantum semigroups admit covariant Markov C ?-dilations," Comm. Math. Phys., 106 (1986), 91–103.
J. Hellmich, Quantenstochastische Integration in Hilbertmoduln, Ph.D. thesis, Tübingen, 2001.
J. Hellmich, C. Köstler, and B. Kümmerer, "Stationary quantum Markov processes as solutions of stochastic differential equations," in: Quantum Probability Banach Center Publications (R. Alicki, M. Bozejko, and W. A. Majewski, editors), vol. 43, Polish Acad. Sci., Institute Math., Warsaw, 1998, pp. 217–229.
C. Köstler, Quanten-Markoff-Prozesse und Quanten-Brownsche Bewegungen, Ph.D. thesis, Tübingen, 2000.
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Skeide, M. Dilations, Product Systems, and Weak Dilations. Mathematical Notes 71, 836–843 (2002). https://doi.org/10.1023/A:1015829130671
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DOI: https://doi.org/10.1023/A:1015829130671