Linnell, P., Lück, W. and Schick, T.: The Ore condition, affiliated operators, and the lamplighter group
- Autor: Peter Linnell, Wolfgang Lück and Thomas Schick
- Titel: The
Ore condition, affiliated operators,
and the lamplighter group
(dvi) (pdf)
-
Preprint, arXiv; to appear in Proceedings of ICTP Trieste conference on High dimensional manifold topology 2001
Abstract:
Let G be the wreath product of Z and Z/2, the so called
lamplighter group and k a commutative ring. We show that kG does
not have a classical ring of quotients (i.e. does not satisfy the
Ore condition). This answers a
Kourovka notebook problem.
Assume that kG is contained in a ring R in
which the element 1-x is invertible, with x a generator of
Z considered as subset of G. Then R is not flat over kG. If
k is the field of complex numbers, this applies
in particular to the algebra UG of unbounded operators
affiliated to the group von Neumann algebra of G.
We present two proofs of these results. The second one is due to
Warren Dicks, who, having seen our argument, found a much simpler and
more elementary proof, which at the same time yielded a more general
result than we had originally proved. Nevertheless, we present both
proofs here, in the hope
that the original arguments might be of use in some other context not
yet known to us.
Thomas Schick