Steklov Institute of Mathematics at St.Petersburg

PREPRINT 01/2003

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- $\amalg_1$

This preprint was accepted January 4, 2003
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  We describe a construction of the factor-representations of type
  $\amalg_1$ corresponding to some {\it pair of the dynamical systems}.
  This is a generalization of the classical von Neumann
  construction of the factors as the crossed-product and of the grouppoid
  approach. We present the natural examples of factor-representations
  of this type for which so called  {\it coupling constant could
  be not equal to one} and consequently there are no spatial
  traces. The simplest example of such situation gives a pair
  of discrete subgroups of Heisenberg group (see \cite{F}), which provide
  also the new kind of factor-representations of type $\amalg_1$ of rotation     algebra.
  Another examples come from the theory of the lattices in the groups and
  from the theory of representations of infinite symmetric group.

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