An lᵖ-Version of von Neumann Dimension for Representations of Equivalence Relations

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In our previous paper, "lᵖ-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 lᵖ(G), as well as the multiplication on the non-commutative Lᵖ 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 l²-Betti numbers for equivalence relations in the lᵖ-case, this may shed some light on the conjectured relation between cost and first l²-Betti number.