10,550 research outputs found
Homogeneous manifolds from noncommutative measure spaces
Let M be a finite von Neumann algebra with a faithful trace . In this
paper we study metric geometry of homogeneous spaces O of the unitary group U
of M, endowed with a Finsler quotient metric induced by the p-norms of ,
, . The main results include the following.
The unitary group carries on a rectifiable distance d_p induced by measuring
the length of curves with the p-norm. If we identify O as a quotient of groups,
then there is a natural quotient distance d'_p that metrizes the quotient
topology. On the other hand, the Finsler quotient metric defined in O provides
a way to measure curves, and therefore, there is an associated rectifiable
distance d_{O,p}. For , we prove that the distances d'_p and d_{O,p}
coincide. Based on this fact, we show that the metric space (O,d'_p) is a
complete path metric space. The other problem treated in this article is the
existence of metric geodesics, or curves of minimal length, in O. We give two
abstract partial results in this direction. The first concerns the initial
values problem and the second the fixed endpoints problem. We show how these
results apply to several examples. In the process, we improve some results
about the metric geometry of U with the p-norm.Comment: 30 pages. The examples in section 4 have been removed, those of
section 5 have been cut dow
Property (T) and rigidity for actions on Banach spaces
We study property (T) and the fixed point property for actions on and
other Banach spaces. We show that property (T) holds when is replaced by
(and even a subspace/quotient of ), and that in fact it is
independent of . We show that the fixed point property for
follows from property (T) when 1
. For simple Lie groups and their lattices, we prove that the fixed point property for holds for any if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive Banach spaces.Comment: Many minor improvement
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