Homomorphisms of convolution algebras

Homomorphisms of Fourier–Stieltjes algebras

“…have been characterized for arbitrary locally compact groups [14,35], and studying the positive and contractive homomorphisms ϕ :…”

“…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].…”

Commuting contractive idempotents in measure algebras

“…Then the one point compactification Ω K,ρ ⊔ {∞} (respectively, topological coproduct, if G K,ρ is compact) is homeomorphic to Γ ρm K ∪ {0}. Indeed, consider the semigroup homomorphism on (T × G K,ρ ) ⊔ {∞} given by (z, g) → zδ g * (ρm K ), ∞ → 0, which has kernel {(ρ(k), k) : k ∈ K} at the identity -a fact which we shall take for granted, thanks to arguments in [9,2]. It suffices to verify that this semigroup homorphism is continuous and that Γ ρm K ∪ {0} is weak*-compact.…”

“…We close with a study of certain groups of measures identified by Greenleaf [2] and Stokke [9] whose identities are contractive idempotents. 0.1.…”

Norm-multiplicative homomorphisms of Beurling algebras

“…This problem was completely solved in the abelian case by Paul Cohen in [1], but remains open in general. Contractive homomorphisms ϕ : L 1 (F ) → M r (G) were first described by Greenleaf in [12], and a characterization of such homomorphisms via a Cohen-type factorization is found in [25]; see [25] for more about the history of this old problem. There had been no known description of the positive (order-preserving) homomorphisms ϕ : L 1 (F ) → M r (G).…”