International audienc

Let L(H) be the algebra of all bounded operators in a Hilbert space H, θ : G → L(H) denotes a strongly continuous unitary representation of a locally compact and second countable group G in H, σ(θ(g)) the spectrum of θ(g) and Conv(S) the convex hull of any subset S in a vector space. We prove here that θ is uniformly continuous if and only if {g ∈ G/0 / ∈ Conv(σ(θ(g)))} is non-meager.Mathematics Subject Classification (2010). 47A10, 47D03.

Let L(X) be the algebra of all bounded operators on a Banach space X, and let θ : G → L(X) be a strongly continuous representation of a locally compact and second countable abelian group G on X. Set σ 1 (θ(g)) := {λ/|λ| | λ ∈ σ(θ(g))}, where σ(θ(g)) is the spectrum of θ(g), and let Σ θ be the set of all g ∈ G such that σ 1 (θ(g)) does not contain any regular polygon of T (by a regular polygon we mean the image under a rotation of a closed subgroup of the unit circle T different from {1}). We prove that θ is uniformly continuous if and only if Σ θ is a non-null set for the Haar measure on G.

In the first part of the paper, some criteria of continuity of representations of a Polish group in a Banach algebra are given. The second part uses the result of the first part to deduce automatic continuity results of Baire morphisms from Polish groups to locally compact groups or unitary groups. In the final part, the spectrum of an element in the range of a strongly but not norm continuous representation is described.