Abstract
Deformations of associative algebras in which time is the deformation parameter are constructed using quantum stochastic flows in which the usual multiplicativity requirement is replaced by multiplicativity with respect to the deformed multiplication. The theory is restricted by a commutativity requirement on the structure maps of the flow, but examples which can be constructed in this way include the noncommutative torus and the Weyl–Moyal deformation.
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References
Belavkin, V. P.: Quantum stochastic positive evolutions: characterization, construction, dilation, Comm. Math. Phys. 184 (1997), 533–566.
Belavkin, V. P.: A new form and *-algebraic structure of quantum stochastic calculus, Rend. Sem. Mat. Fis. Milano 58 (1988), 177–193.
Cohen P. B. and Hudson, R. L.: Generators of quantum stochastic flows and cyclic cohomology, Math. Proc. Cambridge Philos. Soc. 123 (1998), 345–363.
Evans, M. P.: Existence of quantum diffusions, Probab. Theory Related Fields 81 (1989), 473–483.
Eyre, T.M. W.: Quantum Stochastic Calculus and Representations of Lie Superalgebras, Lecture Notes in Math. 1692, Springer, New York, 1998.
Eyre T. M. W. and Hudson, R. L.: Representations of Lie super-algebras and generalized Boson Fermion equivalence in quantum stochastic calculus, Comm. Math. Phys. 186 (1997), 87–94.
Giaquinto A. and Zhang, J. J.: Bialgebra actions, twists and universal deformation formulas, J. Pure Appl. Algebra 128 (1998), 133–151.
Goswami, D. Lindsay, J. M. Sinha K. B. and Wills, S. J.: Dilation ofMarkovian cocycles on a von Neumann algebra, Nottingham University preprint.
Hudson, R. L.: Algebraic theory of quantum diffusions, In: A. Truman (ed.), Stochastic Mechanics and Applications, Proceedings, Swansea, 1986, Lecture Notes in Math. 1325, Springer, New York, 1988, pp. 113–124.
Hudson, R. L.: Quantum stochastic calculus, evolutions and flows, In: Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, 1996, pp. 81–91.
Hudson, R. L. Lindsay J.M. and Parthasarathy, K. R.: Flows of quantum noise, J. Appl. Anal. 4 (1998), 143–160.
Hudson R. L. and Parthasarathy, K. R.: Quantum Itô formula and stochastic evolutions, Comm. Math. Phys. 93 (1984), 301–323.
Hudson R. L. and Robinson, P.M.: Quantum diffusions and the non-commutative torus, Lett. Math. Phys. 15 (1988), 47–53.
Lindsay J. M. and Parthasarathy, K. R.: On the generators of quantum stochastic flows, J. Funct. Anal. 158 (1998), 521–541.
Parthasarathy, K. R.: An Introduction to Quantum Stochastic Calculus, Birkhäuser, Basel, 1992.
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Hudson, R.L., Parthasarathy, K.R. Deformations of Algebras Constructed using Quantum Stochastic Calculus. Letters in Mathematical Physics 50, 115–133 (1999). https://doi.org/10.1023/A:1007604508064
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DOI: https://doi.org/10.1023/A:1007604508064