Constructing alternating 2-cocycles on Fourier algebras

TL;DR

This paper constructs the first examples of non-zero, alternating 2-cocycles on Fourier algebras of certain groups, highlighting the role of operator space structures and technical estimates in the process.

Contribution

It introduces the first known examples of such 2-cocycles on Fourier algebras, utilizing operator space structures and new tensor product inclusion results.

Findings

Existence of non-zero, alternating 2-cocycles on Fourier algebras of some groups

Operator space structure is crucial for the construction

Technical estimates enable derivations with specific boundedness properties

Abstract

Building on recent progress in constructing derivations on Fourier algebras, we provide the first examples of locally compact groups whose Fourier algebras support non-zero, alternating 2-cocycles; this is the first step in a larger project. Although such 2-cocycles can never be completely bounded, the operator space structure on the Fourier algebra plays a crucial role in our construction, as does the opposite operator space structure. Our construction has two main technical ingredients: we observe that certain estimates from [H. H. Lee, J. Ludwig, E. Samei, N. Spronk, Weak amenability of Fourier algebras and local synthesis of the anti-diagonal, Adv. Math., 292 (2016); arXiv 1502.05214] yield derivations that are "co-completely bounded" as maps from various Fourier algebras to their duals; and we establish a twisted inclusion result for certain operator space tensor products, which may be of independent interest.