Realization of compact spaces as cb-Helson sets

We show that, given a compact Hausdorff space $\Omega$, there is a compact group ${\mathbb G}$ and a homeomorphic embedding of $\Omega$ into ${\mathbb G}$, such that the restriction map ${\rm A}({\mathbb G})\to C(\Omega)$ is a complete quotient map of operator spaces. In particular, this shows that there exist compact groups which contain infinite cb-Helson subsets, answering a question raised in [Choi--Samei, Proc. AMS 2013; cf. http://arxiv.org/abs/1104.2953]. A negative result from the same paper is also improved.Comment: v2: AMS-LaTeX, 12 pages. Changes to v1: material on continuity of product representations has been streamlined; other minor changes/corrections made, following referee's recommendations. Accepted by Ann. Funct. Anal; this is, modulo formatting, the author-accepted manuscript (CC-BY licence

Simplicial homology and Hochschild cohomology of Banach semilattice algebras

The ${\ell}^1$-convolution algebra of a semilattice is known to have trivial cohom ology in degrees 1,2 and 3 whenever the coefficient bimodule is symmetric. We ex tend this result to all cohomology groups of degree $\geq 1$ with symmetric coef ficients. Our techniques prove a stronger splitting result, namely that the spli tting can be made natural with respect to the underlying semilattice.Comment: 17pp, preprint version (revised 2006). Final version to appear in Glasgow Math. Journal (2006

Hochschild homology and cohomology of {\ell}^1({\mathbb Z}_+^k)

Building on the recent determination of the simplicial cohomology groups of the convolution algebra ${\ell}^1({\mathbb Z}_+^k)$ [Gourdeau, Lykova, White, 2005] we investigate what can be said for cohomology of this algebra with more general symmetric coefficients. Our approach leads us to a discussion of Harrison homology and cohomology in the context of Banach algebras, and a development of some of its basic features. As an application of our techniques we reprove some known results on second-degree cohomology.Comment: v2: 36 pages, submitted. Abstract added, MSC-classes updated, typos corrected. Presentation trimmed a little. v3: 36 pages; further typos caught, added hyperlinks. Uses Paul Taylor's diagrams.sty macros. Leaner and rearranged version to appear in Q. J. Math. (Oxford

Simplicial cohomology of augmentation ideals in ℓ1(G){\ell}^1(G)

Let $G$ be a discrete group. We give a decomposition theorem for the Hochschild cohomology of $\ell^1(G)$ with coefficients in certain $G$-modules. Using this we show that if $G$ is commutative-transitive, the canonical inclusion of bounded cohomology of $G$ into simplicial cohomology of $\ell^1(G)$ is an isomorphism.Comment: 14pp, uses Paul Taylor's diagrams.sty macros. v3: typos caught and some minor corrections/clarifications of terminology. This is not the final version, which will appear in Proc. Edinburgh Math. So

A gap theorem for the ZL-amenability constant of a finite group

It was shown in [A. Azimifard, E. Samei, N. Spronk, JFA 2009; arxiv 0805.3685] that the ZL-amenability constant of a finite group is always at least 1, with equality if and only if the group is abelian. It was also shown in the same paper that for any finite non-abelian group this invariant is at least 301/300, but the proof relies crucially on a deep result of D. A. Rider on norms of central idempotents in group algebras. Here we show that if G is finite and non-abelian then its ZL-amenability constant is at least 7/4, which is known to be best possible. We avoid use of Rider's result, by analyzing the cases where G is just non-abelian, using calculations from [M. Alaghmandan, Y. Choi, E. Samei, CMB 2014; arxiv 1302.1929], and establishing a new estimate for groups with trivial centre.Comment: v2: AMS-LaTeX 10pt, 20 pages, 1 figure. Some typos corrected and remarks trimmed; new reference added. Final version, to appear in Int. J. Group Theor

Splitting maps and norm bounds for the cyclic cohomology of biflat Banach algebras

We revisit the old result that biflat Banach algebras have the same cyclic cohomology as $\mathbb C$, and obtain a quantitative variant (which is needed in forthcoming joint work of the author). Our approach does not rely on the Connes-Tsygan exact sequence, but is motivated strongly by its construction as found in [Connes,1985] and [Helemskii,1992].Comment: v2: 17 pp LaTeX, slight change to title; some references added and some background motivation expanded upon. Accepted by conference proceedings

Weak and cyclic amenability for Fourier algebras of connected Lie groups

Using techniques of non-abelian harmonic analysis, we construct an explicit, non-zero cyclic derivation on the Fourier algebra of the real $ax+b$ group. In particular this provides the first proof that this algebra is not weakly amenable. Using the structure theory of Lie groups, we deduce that the Fourier algebras of connected, semisimple Lie groups also support non-zero, cyclic derivations and are likewise not weakly amenable. Our results complement earlier work of Johnson (JLMS, 1994), Plymen (unpublished note) and Forrest--Samei--Spronk (IUMJ 2009). As an additional illustration of our techniques, we construct an explicit, non-zero cyclic derivation on the Fourier algebra of the reduced Heisenberg group, providing the first example of a connected nilpotent group whose Fourier algebra is not weakly amenable.Comment: v4: AMS-LaTeX, 26 pages. Final version, to appear in JFA. Includes an authors' correction added at proof stag

Quotients of Fourier algebras, and representations which are not completely bounded

We observe that for a large class of non-amenable groups $G$, one can find bounded representations of $A(G)$ on Hilbert space which are not completely bounded. We also consider restriction algebras obtained from $A(G)$, equipped with the natural operator space structure, and ask whether such algebras can be completely isomorphic to operator algebras; partial results are obtained, using a modified notion of Helson set which takes account of operator space structure. In particular, we show that if $G$ is virtually abelian, then the restriction algebra $A_G(E)$ is completely isomorphic to an operator algebra if and only if $E$ is finite.Comment: v3: 10 pages, minor edits and slight change to title from v2. Final version, to appear in Proc. Amer. Math. So