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.
Categories: math.FA math.GR math.OA