The spine of a Fourier-Stieltjes algebra

Abstract

Abstract. We define the spine A * (G) of the Fourier-Stieltjes algebra B(G) of a locally compact group G. This algebra encodes information about much of the fine structure of B(G); particularly information about certain homomorphisms and idempotents.We show that A * (G) is graded over a certain semi-lattice, that of nonquotient locally precompact topologies on G. We compute the spine's spectrum G * , which admits a semi-group structure. We discuss homomorphisms from A * (G) to B(H) where H is another locally compact group; and we show that A * (G) contains the image of every completely bounded homomorphism from the Fourier algebra A(H) of any amenable group H. We also show that A * (G) contains all of the idempotents in B(G). Finally, we compute examples for vector groups, abelian lattices, minimally almost periodic groups and the ax + b-group; and we explore the complexity of A

Second duals of measure algebras

“…However, in the case where G = G 1 × G 2 , where G 1 and G 2 are compact, infinite groups, the topology formed by taking the product of the given topology on G 1 and the discrete topology on G 2 has the specified properties. This question is related to that of the 'spine' of the algebra M (G); see [54,55].…”

On the structure of invertible elements in certain Fourier–Stieltjes algebras

“…Moreover, elementary arguments based on the Arens-Royden theorem immediately allow for a deeper analysis of invertible elements in motion groups and the ax + b group. We note that these examples cover those for which [IS07] has constructed the spine.…”

“…Here, G z is a locally compact group about the critical point z. On the other hand, in [IS07], Ilie and Spronk develop the non-commutative dual analogue of the spine of an abelian measure algebra and [IS07, Theorem 5.1] provides an explicit description of the above topologies σ i . Moreover, they note that it would be interesting to determine if invertible elements u in B(G) are of the form…”

Homomorphisms of Fourier–Stieltjes algebras

“…After much effort, both versions of the problem were solved in the abelian case by Paul Cohen in 1960 [6,28]. For non-abelian groups, both versions of the problem-which are distinct in the non-abelian case-have been broadly studied by many authors, e.g., see [14,38,16,17,18,27,35,36], but in full generality remain open. In particular, Ilie and Spronk successfully generalized Cohen's results by describing all completely positive, completely contractive and completely bounded homomorphisms ϕ : A(G) → B(H) when G is amenable: any such ϕ is determined by a continuous map α : Y → G where Y ∈ Ω(H), the ring of sets generated by the open cosets of H (we write ϕ = j α ), with α a homomorphism and Y a subgroup precisely when ϕ is completely positive; α an affine map and Y a coset precisely when ϕ is completely contractive; α a piecewise-affine map precisely when ϕ is completely bounded [17,Theorem 3.7].…”

“…To quote E. Kaniuth and A. T.-M. Lau, "when G is a non-compact locally compact abelian group, according to common understanding, the spectrum of B(G) = M ( G) is an intractable object" [21,Section 2.9]. Nevertheless, the structure of ∆(B(G)) and ∆(A) for certain subalgebras A of B(G) has been successfully studied by various authors, e.g., [8,39,24,18], and shown to be fairly accessible in some-typically non-abelian-cases. Results from [39] and [18], and the work herein, allow us to address our main questions, which are as follows:…”