An lp-version of von Neumann dimension for Banach space representations of sofic groups

In [5], A. Gournay defined a notion of lp-dimension for Γ-invariant subspaces of lq(Γ)⊕n, with Γ amenable. The number dimlqlp(Γ,V) is dim V when p=q, and is preserved by a certain class of Γ-equivariant bounded linear isomorphisms. We develop a notion of dimlp,Σ(Y,Γ) where Y is a Banach space with a uniformly bounded action of a sofic group Γ and Σ is a sofic approximation. In particular, our definition makes sense for a large class of non-amenable groups. We also develop a notion of dimSp,Σ(Y,Γ) with Γ an Rω-embeddable group and Sp the space of finite-dimensional Schatten p-class operators. These numbers are invariant under bounded Γ-equivariant linear isomorphisms and under the natural translation action of Γ, dimlp(lp(Γ,V),Γ)=dimV, and dimSp(lp(Γ,V),Γ)=dimV for 1⩽p⩽2. In particular, this shows that lp(Γ,V) is not isomorphic to lp(Γ,W) as a representation of Γ if dimV≠dimW, and Γ is Rω-embeddable. We discuss other concrete computations in a follow-up paper, including proving that our dimension agrees with von Neumann dimension for representations contained in a multiple of the left-regular representation.