Invariants for E_0-semigroups on II_1 factors

Margetts, Oliver and Srinivasan, R. (2013) Invariants for E_0-semigroups on II_1 factors. Communications in Mathematical Physics. ISSN 0010-3616

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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 \emph{super product system} and a $C^*$-semiflow. Using noncommutative It\^o integrals we show that the dimension of the gauge group can be computed from the structure of the \emph{additive cocycles}. We do this for the Clifford flows and even Clifford flows on the hyperfinite \twoone 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 \en-semigroups.

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