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In this paper, we develop a new approach that allows to identify the Gelfand spectrum of weighted Fourier algebras as a subset of an abstract complexification of the corresponding group for a wide class of groups and weights. This generalizes recent related results of Ghandehari-Lee-Ludwig-Spronk-Turowska [11] about the spectrum of Beurling-Fourier algebras on some Lie groups. In the case of discrete groups we show that the spectrum of Beurling-Fourier algebra is homeomorphic to the group itself. [ABSTRACT FROM AUTHOR]

We show that for a connected Lie group G , its Fourier algebra A ( G ) is weakly amenable only if G is abelian. Our main new idea is to show that weak amenability of A ( G ) implies that the anti-diagonal, Δ ˇ G = { ( g , g − 1 ) : g ∈ G } , is a set of local synthesis for A ( G × G ) . We then show that this cannot happen if G is non-abelian. We conclude for a locally compact group G , that A ( G ) can be weakly amenable only if it contains no closed connected non-abelian Lie subgroups. In particular, for a Lie group G , A ( G ) is weakly amenable if and only if its connected component of the identity G e is abelian. [ABSTRACT FROM AUTHOR]

Motivated by the beautiful work of M.A. Rieffel (1965) and of M.E. Walter (1974), we obtain characterisations of the Fourier algebra A ( G ) of a locally compact group G in terms of the class of F -algebras (i.e. a Banach algebra A such that its dual A ′ is a W ⁎ -algebra whose identity is multiplicative on A ). For example, we show that the Fourier algebras are precisely those commutative semisimple F -algebras that are Tauberian, contain a nonzero real element, and possess a dual semigroup that acts transitively on their spectrums. Our characterisations fall into three flavours, where the first one will be the basis of the other two. The first flavour also implies a simple characterisation of when the predual of a Hopf–von Neumann algebra is the Fourier algebra of a locally compact group. We also obtain similar characterisations of the Fourier–Stieltjes algebras of G . En route, we prove some new results on the problem of when a subalgebra of A ( G ) is the whole algebra and on representations of discrete groups. [ABSTRACT FROM AUTHOR]

Abstract: This paper is an invitation to the study of the Fourier–Stieltjes algebra , the linear span of the continuous positive definite complex-valued functions on a topological group G. We study , when G has a host algebra or a group -algebra, the analogue of the group -algebra of a locally compact group. Our main challenge is that a topological group G cannot have a positive regular Borel measure which is left translation invariant unless G is locally compact. [Copyright &y& Elsevier]

We investigate Beurling-Fourier algebras, a weighted version of Fourier algebras, on various Lie groups focusing on their spectral analysis. We will introduce a refined general definition of weights on the dual of locally compact groups and their associated Beurling-Fourier algebras. Constructions of nontrivial weights will be presented focusing on the cases of representative examples of Lie groups, namely S U (n) , the Heisenberg group H , the reduced Heisenberg group H r , the Euclidean motion group E (2) and its simply connected cover E ˜ (2). We will determine the spectrum of Beurling-Fourier algebras on each of the aforementioned groups emphasizing its connection to the complexification of underlying Lie groups. We also demonstrate that "polynomially growing" weights do not change the spectrum and show the associated regularity of the resulting Beurling-Fourier algebras. [ABSTRACT FROM AUTHOR]

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 [12] 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. [ABSTRACT FROM AUTHOR]

We introduce the notion of a generalised F-algebra as a Banach algebra A whose Banach space dual is a J B W ⁎ -triple with a certain additional condition on a distinguished character of A. This generalises the well-known notion of F -algebras, which in turn, includes as special cases many important classes of Banach algebras on locally compact groups and semigroups. Our main use of this new notion is to give some axiomatic characterisations of the Fourier algebras on locally compact groups, strengthening our previous work [37]. But we also quickly review some well-known results in F -algebras that can be extended to the context of generalised F -algebras. [ABSTRACT FROM AUTHOR]