Sets of p-multiplicity in locally compact groups

Abstract

Abstract. We initiate the study of sets of p-multiplicity in locally compact groups and their operator versions. We show that a closed subset E of a second countable locally compact group G is a set of p-multiplicity if and only if the set E * = {(s, t) : ts −1 ∈ E} is a set of operator p-multiplicity. We exhibit examples of sets of p-multiplicity, establish preservation properties for unions and direct products, and prove a pversion of the Stone-von Neumann Theorem.

Herz–Schur multipliers of dynamical systems

“…(i)⇒(ii) Given u ∈ M cb A(G), let Ψ u be the weak* continuous completely bounded map on B(L 2 (G)) corresponding to the Schur multiplier N (u) (see Remark 3.9). We claim that (41) (…”

“…If u ∈ J then S θ u (T ) = T and hence, by (41) and the fact that Φ commutes with S θ u , we have Ψ u (L ω (Φ(T ))) = L ω (Φ(T )). Thus, for every u ∈ J, the operator L ω (Φ(T )) is u-harmonic in the sense of [24].…”

“…In the proof of Theorem 3.18 below, we will need the following improvement of [41,Lemma 3.9]. Lemma 3.16.…”

Reduced spectral synthesis and compact operator synthesis

“…Recall [55] that an ω-closed set κ ⊆ X × Y is called a set of operator p-multiplicity if M max (κ) ∩ C p = {0}. We say that κ ⊆ X × Y is a set of operator p-synthesis if M max (κ) ∩ C p = M min (κ) ∩ C p .…”

Reduced spectral synthesis and compact operator synthesis

“…Using a complex interpolation argument, one can then easily show that Schur multipliers leave the ideal C p invariant. It thus suffices to show that Recall [54] that an ω-closed set κ ⊆ X × Y is called a set of operator p-multiplicity if M max (κ) ∩ C p = {0}. We say that κ ⊆ X × Y is a set of operator p-synthesis if M max (κ) ∩ C p = M min (κ) ∩ C p .…”