We introduce the class of quasi-pure E0-semigroups acting on a von Neumann algebra as that with equal tail and fixed-point algebras and we describe these notable algebras for E0-semigroups of B(H) where H is a separable Hilbert space. We consider the classes of pure, quasi-pure and ergodic E0-semigroups induced by essential states of the spectral C+-algebra of a product system and we characterize spatial product systems in terms of the existence of certain essential states of the corresponding spectral C+-algebra.