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Invariants for E0-Semigroups on II1 Factors

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Abstract

We introduce four new cocycle conjugacy invariants for E 0-semigroups on II 1 factors: a coupling index, a dimension for the gauge group, a super product system and a C*-semiflow. Using noncommutative Itô integrals we show that the dimension of the gauge group can be computed from the structure of the additive cocycles. We do this for the Clifford flows and even Clifford flows on the hyperfinite II 1 factor, and for the free flows on the free group factor \({L(F_\infty)}\) . In all cases the index is 0, which implies they have trivial gauge groups. We compute the super product systems for these families and, using this, we show they have trivial coupling index. Finally, using the C*-semiflow and the boundary representation of Powers and Alevras, we show that the families of Clifford flows and even Clifford flows contain infinitely many mutually non-cocycle-conjugate E 0-semigroups.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster, LA1 4YF, U.K

    Oliver T. Margetts

  2. Department of Mathematics, Chennai Mathematical Institute, H1, SIPCOT IT Park, Kelambakkam, Siruseri, 603103, India

    R. Srinivasan

Authors
  1. Oliver T. Margetts
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  2. R. Srinivasan
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Correspondence to R. Srinivasan.

Additional information

Communicated by Y. Kawahigashi

We humbly dedicate this paper to the memory of Bill Arveson.

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Margetts, O.T., Srinivasan, R. Invariants for E0-Semigroups on II1 Factors. Commun. Math. Phys. 323, 1155–1184 (2013). https://doi.org/10.1007/s00220-013-1790-2

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