Open Access

Permutations in tensor products of factors, and $L^{2}$ cohomology of configuration spaces

Abstract

We prove that the natural action of permutations in a tensor product of type $\mathrm{II}$ factors is free, and compute the von Neumann trace of the projection onto the space of symmetric and antisymmetric elements respectively. We apply this result to computation of von Neumann dimensions of the spaces of square-integrable harmonic forms ($L^{2}$-Betti numbers) of $N$-point configurations in Riemannian manifolds with infinite discrete groups of isometries.

Article Information

 Title Permutations in tensor products of factors, and $L^{2}$ cohomology of configuration spaces Source Methods Funct. Anal. Topology, Vol. 12 (2006), no. 4, 341-352 MathSciNet MR2279871 Copyright The Author(s) 2006 (CC BY-SA)

Authors Information

Alexei Daletskii
Nottingham Trent University, Nottingham, UK

Alexander Kalyuzhnyi
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine

Citation Example

Alexei Daletskii and Alexander Kalyuzhnyi, Permutations in tensor products of factors, and $L^{2}$ cohomology of configuration spaces, Methods Funct. Anal. Topology 12 (2006), no. 4, 341-352.

BibTex

@article {MFAT383,
AUTHOR = {Daletskii, Alexei and Kalyuzhnyi, Alexander},
TITLE = {Permutations in tensor products of factors, and $L^{2}$ cohomology of configuration spaces},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {12},
YEAR = {2006},
NUMBER = {4},
PAGES = {341-352},
ISSN = {1029-3531},
URL = {http://mfat.imath.kiev.ua/article/?id=383},
}