Kyoto Journal of Mathematics

Cohomology for spatial superproduct systems

Oliver T. Margetts and R. Srinivasan

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Abstract

We introduce a cohomology theory for spatial superproduct systems and compute the 2-cocycles for some basic examples called Clifford superproduct systems, thereby distinguishing them up to isomorphism. This consequently proves that a family of E0-semigroups on type III factors, which we call CAR flows, are noncocycle-conjugate for different ranks. Similar results follow for the even CAR flows as well. We also compute the automorphism group of the Clifford superproduct systems.

Article information

Source
Kyoto J. Math., Volume 59, Number 1 (2019), 53-75.

Dates
Received: 2 August 2016
Revised: 26 December 2016
Accepted: 28 December 2016
First available in Project Euclid: 23 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1534989637

Digital Object Identifier
doi:10.1215/21562261-2018-0002

Mathematical Reviews number (MathSciNet)
MR3934623

Zentralblatt MATH identifier
07081622

Subjects
Primary: 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]
Secondary: 46L40: Automorphisms 46L53: Noncommutative probability and statistics 46C99: None of the above, but in this section

Keywords
∗-endomorphisms $E_{0}$-semigroups type III factors noncommutative probability superproduct systems

Citation

Margetts, Oliver T.; Srinivasan, R. Cohomology for spatial superproduct systems. Kyoto J. Math. 59 (2019), no. 1, 53--75. doi:10.1215/21562261-2018-0002. https://projecteuclid.org/euclid.kjm/1534989637


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References

  • [1] G. G. Amosov, On cocycle conjugacy of quasifree endomorphism semigroups on the CAR algebra, J. Math. Sci. (N.Y.) 105 (2001), 2496–2503.
  • [2] H. Araki and W. Wyss, Representations of canonical anticommutation relations, Helv. Phys. Acta 37 (1964), 136–159.
  • [3] W. Arveson, Noncommutative Dynamics and $E$-Semigroups, Springer Monogr. Math., Springer, New York, 2003.
  • [4] P. Bikram, CAR flows on type III factors and their extendability, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 17 (2014), art. ID 1450026.
  • [5] P. Bikram, M. Izumi, R. Srinivasan, and V. S. Sunder, On extendability of endomorphisms and of $\mathrm{E}_{0}$-semigroups on factors, Kyushu J. Math. 68 (2014), 165–179.
  • [6] M. Izumi and R. Srinivasan, Generalized CCR flows, Comm. Math. Phys. 281 (2008), 529–571.
  • [7] M. Izumi and R. Srinivasan, Toeplitz CAR flows and type I factorizations, Kyoto J. Math. 50 (2010), 1–32.
  • [8] V. Liebscher, Random sets and invariants for (type II) continuous tensor product systems of Hilbert spaces, Mem. Amer. Math. Soc. 199 (2009), no. 930.
  • [9] O. Margetts and R. Srinivasan, Invariants for ${E}_{0}$-semigroups on II$_{1}$ factors, Comm. Math. Phys. 323 (2013), 1155–1184.
  • [10] O. Margetts and R. Srinivasan, Invariants for ${E}_{0}$-semigroups on II$_{1}$ factors, preprint, arXiv:1209.1283v1 [math.OA].
  • [11] O. Margetts and R. Srinivasan, Non-cocycle-conjugate ${E}_{0}$-semigroups on factors, preprint, arXiv:1404.5934v2 [math.OA].
  • [12] R. T. Powers, A nonspatial continuous semigroup of $*$-endomorphisms of $B(H)$, Publ. Res. Inst. Math. Sci. 23 (1987), 1053–1069.
  • [13] R. T. Powers, An index theory for semigroups of $*$-endomorphisms of $B(H)$ and type II$_{1}$ factors, Canad. J. Math. 40 (1988), 86–114.
  • [14] R. T. Powers and E. Størmer, Free states of the canonical anticommutation relations, Comm. Math. Phys. 16 (1970), 1–33.
  • [15] B. V. Rajarama Bhat and R. Srinivasan, On product systems arising from sum systems, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8 (2005), 1–31.
  • [16] M. Takesaki, Theory of Operator Algebras, II, Encyclopaedia Math. Sci. 125, Springer, Berlin, 2003.
  • [17] B. Tsirelson, “Non-isomorphic product systems” in Advances in Quantum Dynamics (South Hadley, MA, 2002), Contemp. Math. 335, Amer. Math. Soc., Providence, 2003, 273–328.