Journal of Operator Theory

Volume 51, Issue 1, Winter 2004 pp. 161-179.

One parameter semigroups of endomorphisms of factors of type II$_1$

Authors: Alexis Alevras
Author institution: Department of Mathematics, University of California, Santa Barbara, CA 93106, USA

Summary: We study invariants for continuous semigroups of $\star$-endo\-mor\-phisms of type {\rm II}$_1$-factors. An index is defined, based on R. Powers's notion of the boundary representation, and computed for all known examples on the hyperfinite {\rm II}$_1$-factor ${\cal R}$, as well as for examples on $L(F_{\infty})$. We also introduce the analogue of W. Arveson's continuous tensor product system associated with an $E_0$-semigroup, and show that it is a complete invariant under cocycle conjugacy.

Keywords: $*$-endomorphisms, $E_0$-semigroups, Hilbert modules, Product systems, $II_1$-factors

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