Herz–Schur multipliers of dynamical systems

Weak amenability for dynamical systems

“…In this section we review the definitions and results of [13] required later, as well as establish notation. Throughout, G will denote a second-countable, locally compact, topological group, endowed with left Haar measure m; integration on G, with respect to m, over the variable s is simply denoted ds.…”

“…In [13] the present author, with Todorov and Turowska, introduced and studied Herz-Schur multipliers of a C * -dynamical system, extending the classical notion of a Herz-Schur multiplier (see De Cannière-Haagerup [6]). We now recall the definitions and results needed here; the classical definitions of Herz-Schur multipliers are the special case A = C of the definitions below.…”

“…t , a ∈ A, t ∈ G, has an extension to a completely bounded weak * -continuous map on A w * α,θ G. Observe that [13,Remark 3.4] shows that Herz-Schur θ-multipliers of (A, G, α) act in the same way as Herz-Schur (A, G, α)-multipliers, when viewed through a weak * -continuous functional. To simplify notation we will often omit the superscript θ from the maps S F associated to the multipliers defined above; it will be clear from the presence/absence of θ elsewhere in the notation where S F is acting.…”

Operator System Structures and Extensions of Schur Multipliers

“…Identity (12) easily implies that there exists a (normal) map Ψ : S → B(K) such that Φ = Ψ (n) . Since Φ is hermitian and contractive, so is Ψ.…”

Positive Herz–Schur multipliers and approximation properties of crossed products

“…Hence, by Theorem 2.1, N (h F ) is a Schur A-multiplier with N (h F ) m ≤ |F | Φ cb . By [MTT,Theorem 3.8], h F is a Herz-Schur (A, G, α)-multiplier. If Φ is completely positive then we can choose V p = W p and hence V (s) = W (s) for every s ∈ G. In this case, by Theorem 2.6 and Theorem 2.8, h F is a completely positive Herz-Schur (A, G, α)-multiplier.…”