## Constructing alternating 2-cocycles on Fourier algebras

#### Authors

Yemon Choi

* Corresponding author

Keywords: alternating cocycle, co-completely bounded, Fourier algebra, Hochschild cohomology, operator space, opposite operator space, tensor product.

MSC 2020: 16E40, 46J10, 46L07 (primary); 43A30, 46M05 (secondary)

DOI: 10.1016/j.aim.2021.107747

#### Status

To appear as Adv. Math. 385 (2021) article 107747 (28 pages).

Preprint version available at arXiv 2008.02226

#### Reviews

[ MR 4246974 (pending) | Zbl 07358492 (pending) ]

#### 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.