An l^{p}-Version of von Neumann Dimension for Representations of Equivalence Relations
Abstract
In our previous paper, "l^{p}-Version of von Neumann Dimension for Banach Space Representations of Sofic Groups," we define an extended version of von Neumann dimension for actions of a sofic group on a Banach space. This dimension was studied especially for the translation action of G on l^{p}(G), as well as the multiplication on the non-commutative L^{p} space associated with the group von Neumann algebra. We discuss how one can similarly define an extended dimension for representations of an equivalence relation. We also define an analogue of l^{2}-Betti numbers for equivalence relations in the l^{p}-case, this may shed some light on the conjectured relation between cost and first l^{2}-Betti number.
Keywords
Cite
@article{arxiv.1302.2293,
title = {An l^{p}-Version of von Neumann Dimension for Representations of Equivalence Relations},
author = {Ben Hayes},
journal= {arXiv preprint arXiv:1302.2293},
year = {2013}
}
Comments
45 pages, fixed some minor typographical errors from the first version, add some crucial references that were omitted the first time