An l^{p}-Version of von Neumann Dimension for Banach Space Representation of Sofic Groups II

TL;DR

This paper extends the concept of von Neumann dimension to Banach space representations of sofic groups, specifically defining l^{p}-dimension and proving non-isomorphism of certain group actions, addressing key conjectures from prior work.

Contribution

It introduces an l^{p}-dimension for Banach space representations of sofic groups and proves non-isomorphism results, advancing the understanding of these representations.

Findings

Defined l^{p}-dimension for Banach space representations

Proved actions on l^{p}(G)^{⊕n} are pairwise non-isomorphic

Addressed conjectures from previous work

Abstract

In previous work, we defined extended versions of von Neumann dimension for Banach space representations of sofic groups. The main application was a definition of l^{p}-dimension and, using this, a proof that the actions of a countable discrete group G on l^{p}(G)^{oplus n} are pairwise non-isomorphic. We answer most of the Conjectures stated at the end of the previous paper "An l^{p}-Version of von Neumann Dimension for Banach Space Representation of Sofic Groups." Tackling the case of finte-dimensional representations involved passing to equivalence relations. This is part of current research to be explained in the paper "An l^{p}-Version of von Neumann Dimension for Representations of Equivalence Relations."