Spectral representations for abelian automorphism groups of von Neumann algebras

Abstract

Let A be a von Neumann algebra, let σ be a strongly continuous representation of the locally compact abelian group G as ∗-automorphisms of A . Let M( σ) be the Banach algebra of bounded linear operators on A generated by ∝ σ t dμ( t) ( μ ϵ M( G)). Then it is shown that M( σ) is semisimple whenever either (i) A has a σ-invariant faithful, normal, semifinite, weight (ii) σ is an inner representation or (iii) G is discrete and each σ t is inner. It is shown that the Banach algebra L( σ) generated by ∝ ƒ(t)σ t dt (ƒ ϵ L 1(G)) is semisimple if a is an integrable representation. Furthermore, if σ is an inner representation with compact spectrum, it is shown that L( σ) is embedded in a commutative, semisimple, regular Banach algebra with isometric involution that is generated by projections. This algebra is contained in the ultraweakly continuous linear operators on A . Also the spectral subspaces of σ are given in terms of projections.