* 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)
To appear as Adv. Math. 385 (2021) article 107747 (28 pages).
Preprint version available at arXiv 2008.02226
[ MR 4246974 (pending) | Zbl 07358492 (pending) ]
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.
This paper was written between May 2020 and September 2020, but was originally part of a project from 2015-17 that ended up on hold. On resumption, it turned out that various improvements to the results and proofs were possible: the technical estimates used to prove the "twisted inclusion" for operator-space tensor products could be sharpened using the Haagerup tensor norm; and by making systematic use of results scattered throughout the 2016 paper of Lee–Ludwig–Samei–Spronk, I could considerably broaden the scope of examples covered by the main result.
The paper includes an improved version of some material from arXiv 1606.06287v2 which will not be submitted for publication.
The "headline result" of the paper is Theorem 1.1, which says that there are non-zero alternating 2-cocycles on A(G) whenever G=SU(n), SL(n,R) or Isom(Rn) for n≥4. This list of groups is illustrative, rather than comprehensive; the same result is achieved for many other G. However, the results in the paper fall short of proving that such 2-cocycles exist for all non-abelian connected Lie groups G of sufficiently high rank. In particular, the case of G=H×H where H is the 3-dimensional real Heisenberg group remains unresolved.
Remark added 2021/06/17: Something that I had noticed during the "research stage" but forgot to include in the final paper, is that in some but not all cases covered by Theorem 1.1, the 2-cocycles we obtain are not just alternating, but cyclic. This is because the non-zero derivations constructed by Ghandehari and me in our 2014 JFA paper are cyclic, and informally speaking, wedging two cyclic derivations gives an alternating cyclic 2-cocycle. While no immediate applications come to mind, it does show that we obtain explicit non-zero classes in (continuous) cyclic cohomology for certain Fourier algebras.