Generalized CCR Flows
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We introduce a new construction of E 0-semigroups, called generalized CCR flows, with two kinds of descriptions: those arising from sum systems and those arising from pairs of C 0-semigroups. We get a new necessary and sufficient condition for them to be of type III, when the associated sum system is of finite index. Using this criterion, we construct examples of type III E 0-semigroups, which can not be distinguished from E 0-semigroups of type I by the invariants introduced by Boris Tsirelson. Finally, by considering the local von Neumann algebras, and by associating a type III factor to a given type III E 0-semigroup, we show that there exist uncountably many type III E 0-semigroups in this family, which are mutually non-cocycle conjugate.
KeywordsProduct System Toeplitz Operator Real Hilbert Space Spectral Density Function Complex Hilbert Space
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- 5.Arveson, W.: Non-commutative dynamics and E-semigroups. Springer Monograph in Math, Berlin-Heidelberg-New York: Springer, 2003Google Scholar
- 8.Feller, W.: An Introduction to Probability Theory and its Applications. Vol. II. Second edition, New York-London-Sydney: John Wiley & Sons, Inc., 1971Google Scholar
- 11.Koosis, P.: Introduction to H p Spaces. Second edition. With two appendices by V. P. Havin. Cambridge Tracts in Mathematics, 115. Cambridge: Cambridge University Press, 1998Google Scholar
- 12.Liebscher, V.: Random sets and invariants for (type II) continuous product systems of Hilbert spaces, http://arxiv.org/list/math.PR/0306365, 2003
- 16.Price, G.L., Baker, B.M., Jorgensen, P.E.T., Muhly, P.S. (eds.),: Advances in Quantum Dynamics, (South Hadley, MA, 2002) Contemp. Math. 335, Providence, RI: Amer. Math. Soc., 2003Google Scholar
- 18.Tsirelson, B.: Non-isomorphic product systems. Advances in Quantum Dynamics (South Hadley, MA, 2002), Contemp. Math., 335, Providence, RI: Amer. Math. Soc., 2003, pp. 273–328Google Scholar
- 20.Yosida, K.: Functional Analysis. Sixth edition. Berlin-New York: Springer-Verlag, 1980Google Scholar
- 21.Zhu, K.H.: Operator Theory in Function Spaces. Monographs and Textbooks in Pure and Applied Mathematics 139. New York: Marcel Dekker, Inc., 1990Google Scholar