Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Every sum system is divisible

Author: Masaki Izumi
Journal: Trans. Amer. Math. Soc. 361 (2009), 4247-4267
MSC (2000): Primary 46L55, 47D03, 81S05
Published electronically: March 13, 2009
MathSciNet review: 2500888
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that every sum system is divisible. Combined with B. V. R. Bhat and R. Srinivasan's result, this shows that every product system arising from a sum system (and every generalized CCR flow) is either of type I or type III. A necessary and sufficient condition for such a product system to be of type I is obtained.

References [Enhancements On Off] (What's this?)

  • 1. H. Araki, On quasifree states of the canonical commutation relations. II. Publ. Res. Inst. Math. Sci. 7 (1971/72), 121-152. MR 0313834 (47:2388)
  • 2. W. Arveson, Continuous analogues of Fock space, Mem. Amer. Math. Soc. 80(409):1-66, 1989. MR 987590 (90f:47061)
  • 3. W. Arveson, Continuous analogues of Fock space. IV. Essential states, Acta Math. 164 (3/4) 265-300, 1990. MR 1049159 (91d:46074)
  • 4. W. Arveson, Noncommutative dynamics and $ E$-semigroups, Springer Monographs in Math, Springer, 2003. MR 1978577 (2004g:46082)
  • 5. B. V. R. Bhat and R. Srinivasan, On product systems arising from sum systems, Infinite dimensional analysis, quantum probability and related topics, Vol. 8, Number 1, March 2005. MR 2126876 (2006e:46075)
  • 6. A. van Daele, Quasi-equivalence of quasi-free states on the Weyl algebra. Comm. Math. Phys. 21 (1971), 171-191. MR 0287844 (44:5046)
  • 7. M. Izumi, A perturbation problem for the shift semigroup. J. Funct. Anal. 251, (2007), 498-545. MR 2356422
  • 8. M. Izumi and R. Srinivasan, Generalized CCR flows. Commun. Math. Phys. 281, (2008), 529-571. MR 2410905
  • 9. V. Liebscher, Random sets and invariants for (type $ II$) continuous product systems of Hilbert spaces, Preprint math.PR/0306365.
  • 10. K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, Birkhäuser Basel, Boston, Berlin (1992). MR 1164866 (93g:81062)
  • 11. R. T. Powers, A nonspatial continuous semigroup of $ *$-endomorphisms of $ {\mathfrak{B}}({\mathfrak{H}})$, Publ. Res. Inst. Math. Sci. 23 (1987), 1053-1069. MR 935715 (89f:46118)
  • 12. M. Skeide, Existence of $ E_0$-semigroups for Arveson systems: Making two proofs into one. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006), 373-378. MR 2256500 (2007e:46057)
  • 13. B. Tsirelson, Non-isomorphic product systems. Advances in Quantum Dynamics (South Hadley, MA, 2002), 273-328, Contemp. Math., 335, Amer. Math. Soc., Providence, RI, 2003. MR 2029632 (2005b:46149)
  • 14. K. Yosida, Functional Analysis. Sixth edition. Springer-Verlag, Berlin-New York, 1980. MR 617913 (82i:46002)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 46L55, 47D03, 81S05

Retrieve articles in all journals with MSC (2000): 46L55, 47D03, 81S05

Additional Information

Masaki Izumi
Affiliation: Department of Mathematics, Kyoto University, Kyoto, Japan

Keywords: $E_0$-semigroups, product system, type I, type III
Received by editor(s): August 14, 2007
Published electronically: March 13, 2009
Additional Notes: This work was supported by JSPS
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society