Completeness of eigenvectors of group representations of operators whose Arveson spectrum is scattered

Abstract

Abstract. We establish the following result.Theorem. Let α : G → L(X) be a σ(X, X * ) integrable bounded group representation whose Arveson spectrum Sp(α) is scattered. Then the subspace generated by all eigenvectors of the dual representation α * is w * dense in X * . Moreover, the σ(X, X * ) closed subalgebra Wα generated by the operators αt (t ∈ G) is semisimple.If, in addition, X does not contain any copy of c 0 , then the subspace spanned by all eigenvectors of α is σ(X, X * ) dense in X. Hence, the representation α is almost periodic whenever it is strongly continuous. Spectral theory for integrable bounded group representationsThroughout this paper G will denote a locally compact abelian (LCA) group with identity e and G will denote the dual group of G. The multiplication on LCA groups will be written by addition. Let L 1 (G) (resp. M(G)) be the usual group algebra (resp. measure algebra) with convolution as product operation. We refer to [11] or [21] for basic knowledge of Harmonic Analysis on LCA groups.Given a complex Banach space X, let L(X) be the Banach algebra of all bounded linear operators on X. Take a LCA group G. A bounded group representation α of G on X is a mapping α : G → L(X) satisfying the following properties:(a) Group property: α e = I X the identity operator on X and α s+t = α s α t for all s, t ∈ G; (b) Boundedness: α := sup t∈G α t < ∞. Moreover, α is called strongly (resp. weakly) continuous if for each x ∈ X the mapping t → α t x is norm (resp. weakly) continuous. We need a further notion. Definition 1.1. A bounded group representation α : G → L(X) of G on X is called integrable if there exists a subspace X * ⊂ X * satisfying the following

Continuous Spectrum of Automorphism Groups and the Infraparticle Problem

“…The spectral subspaces (2.4) and the pure-point spectrum have been thoroughly studied in the literature. (See [Ar82,Pe] for a review of the former subject and [Ba78,Hu99] for interesting results on the latter). However, there does not seem to exist any generally accepted definition of the continuous Arveson spectrum.…”

Spectral Theory of Automorphism Groups and Particle Structures in Quantum Field Theory

“…The subject of Cauchy problems in the theory of differential equations [AB97] includes an interesting line of developments which bears some similarity to the present work: Assume that the spectrum of D is countable. Then, under some additional conditions, the eigenvectors of D span a norm dense subspace in A [Ba78,Hu99]. In this case the representation α is called almost periodic, as all its orbits t → α t (A), A ∈ A, belong to the class AP (R, A) of almost periodic functions with values in A.…”

Spectral theory of automorphism groups and particle structures in quantum field theory

“…The subject of Cauchy problems in the theory of differential equations [AB97] includes an interesting line of developments which bears some similarity to the present work: Assume that the spectrum of D is countable. Then, under some additional conditions, the eigenvectors of D span a norm dense subspace in A [Ba78,Hu99]. In this case the representation α is called almost periodic, as all its orbits t → α t (A), A ∈ A, belong to the class AP (R, A) of almost periodic functions with values in A.…”