Continuity of representations of topological groups
Let L(X) be the algebra of all bounded operators on a Banach space X, and let t:G⇾L(X) be a representation of a topological group G in X. For every element g in the group G, we consider the projection on the unit circle T of the spectrum s(t(g)) of the invertible operator t(g), so we set s1(t(g)):={l/|l|, l∊s(t(g))}, and we consider the set S of all elements g in the group G such that s1(t(g)) does not contain any regular polygon of T, so we set S:={g∊ G / ∄ P∊P' / P ⊆ s1(t(g))}, where P' denotes the set of regular polygons of T (we call regular polygon in T the image by a rotation of a closed subgroup of T different from {1}). In the first part, we set out the principal results and notations subsequently used. When G is a locally compact abelian group, we prove in the second part that t is uniformly continuous if and only if t is measurable (L(X) is equipped with the norm topology) and if more G is second countable and t strongly continuous, we state in the third part that t is uniformly continuous if and only if S is non meager. In the same way, we show that t is uniformly continuous if and only if S is a non null set for the Haar measure on G. When G is a locally compact group and t a unitary representation of G in a Hilbert space H, we show also in the second part that t is uniformly continuous if and only if t is measurable, and if more G is metrizable and t strongly continuous, we prove in the third part that t is uniformly continuous if and only if {g∊ G / 0∉ Conv(s(t(g)))} is non meager, where Conv(S) denotes the convex hull of any subset S in a vector space.