ENERGY REPRESENTATIONS OF INFINITE DIMENSIONAL GAUGE GROUPS IN NONCOMMUTATIVE GEOMETRY

Let M be a compact smooth manifold, let be a unital involutive subalgebra of the von Neumann algebra £ (H) of bounded linear operators of some Hilbert space H, let be the unital involutive algebra , let be an hermitian projective right -module of finite type, and let be the gauge group of unitary elements of the unital involutive algebra of right -linear endomorphisms of . We first prove that noncommutative geometry provides the suitable setting upon which a consistent theory of energy representations can be built. Three series of energy representations are constructed. The first consists of energy representations of the gauge group , being the group of unitary elements of , associated with integrable Riemannian structures of M, and the second series consists of energy representations associated with (d, ∞)-summable K-cycles over . In the case where is a von Neumann algebra of type II₁ a third series is given: we introduce the notion of regular quasi K-cycle, we prove that regular quasi K-cycles over always exist, and that each of them induces an energy representation.