Spectral properties of continuous representations of topological groups

Abstract

International audienc

Unitary representations of groups, continuity and spectrum

“…In [3] the following result was obtained: let T be the unit circle in the complex plane C and P the set of the regular polygons of T (we call polygon the range by some rotation of a closed subgroup of T different from {1}), if θ is a strongly continuous representation of a second countable locally compact abelian group G in a Banach space then, either θ is uniformly continuous or the set Σ θ := {g ∈ G/ there is no P ∈ P/P ⊆ σ 1 (θ(g))} is meager. Here σ 1 (θ(g)) := {λ/|λ|, λ ∈ σ(θ(g))} where σ(θ(g)) denotes the spectrum of θ(g), note that σ 1 (θ(g)) is well defined since θ(g) is invertible for all g ∈ G. So when the representation is not uniformly continuous, the spectrum of θ(g) has a rather dispersed angular distribution except for g in a meager subset of G. This work was motivated by a characterization of the uniform continuity of strongly continuous one-parameter groups (see [7]): if (T (t)) t∈R is such a group acting on a Banach space then (T (t)) t∈R is uniformly continuous if and only if {t ∈ R/σ 1 (T (t)) = T} is non-meager.…”

Representations of Polish groups and continuity

“…Introduction. In [8], [7], [3], [24], [19] various sufficient conditions of continuity for representations of locally compact groups on Banach spaces, or more generally in Banach algebras, are given. In [8], [7], [3] these conditions are of spectral nature.…”

“…In [8], [7], [3], [24], [19] various sufficient conditions of continuity for representations of locally compact groups on Banach spaces, or more generally in Banach algebras, are given. In [8], [7], [3] these conditions are of spectral nature. To be more precise, in [8], it is proved that if G is a locally compact abelian group, A a unital Banach algebra and θ : G → A a locally bounded (norm bounded on compact subsets of G) representation, then the continuity of θ is equivalent to the a priori weaker condition ρ(θ(g) − I) → 0 as g → e where ρ denotes the spectral radius in A and e is the unit of G. (This condition is often called spectral continuity for θ.)…”

“…In the locally compact abelian case [3], [4], we can take Ω to be the Gelfand spectrum  of the (abelian) algebra A. In the case G = R, such theorems can already be found in [26] and in the classical treatise [14].…”

“…In this work, we wish to extend them, whenever possible, to Polish and not necessarily locally compact groups. The aim of Section 2 is to generalize to any Polish group the continuity criteria of [3] and [24].…”