Ergodic states in a non commutative ergodic theory

Abstract

Using the Godement mean ℳ of positive-type functions over a groupG, we study “ℳ-abelian systems” {\(\mathfrak{A}\), α} of aC*-algebra\(\mathfrak{A}\) and a homomorphic mapping α of a groupG into the homomorphism group of\(\mathfrak{A}\). Consideration of the Godement mean off(g)U g withf a positive-type function overG andU a unitary representation ofG first yields a generalized mean-ergodic theorem. We then define the Godement mean off(g) π(α g (A)) withA e\(\mathfrak{A}\) and π a covariant representation of the system {\(\mathfrak{A}\), α} for which theG-invariant Hilbert space vectors are cyclic and study its properties, notably in relation with ergodic and weakly mixing states over\(\mathfrak{A}\). Finally we investigate the “discrete spectrum” of covariant representations of {\(\mathfrak{A}\), α} (i.e. the direct sum of the finite-dimensional subrepresentations of the associated representations ofG).