Roberto Giuntini Antonio Ledda Francesco Paoli A New View of Effects in a Hilbert Space Abstract. We investigate certain Brouwer-Zadeh lattices that serve as abstract counter- parts of lattices of effects in Hilbert spaces under the spectral ordering. These algebras, called PBZ*-lattices, can also be seen as generalisations of orthomodular lattices and are remarkable for the collapse of three notions of “sharpness” that are distinct in general Brouwer-Zadeh lattices. We investigate the structure theory of PBZ*-lattices and their reducts; in particular, we prove some embedding results for PBZ*-lattices and provide an initial description of the lattice of PBZ*-varieties. Keywords: Orthomodular lattice, Brouwer–Zadeh lattice, Kleene lattice, Unsharp quantum theory, Effect, Spectral ordering. 1. Introduction Any physical theory determines a class of event-state systems 〈E , S〉, where E contains the events that may occur relative to a given system, while S contains the states that such a physical system, described by the theory, may assume. In the case of quantum mechanics, and in particular within the framework of the traditional Hilbert space model [17, p. 52 ff.], it is customary to identify E with the set Π (H) of all projection operators of a Hilbert space H (which is in bijective correspondence with the set of all closed subspaces thereof), and S with the set S (H) of density operators of H. It is well-known that Π (H) can be made into the universe of an orthomodular lattice if we endow it with intersection, as well as with the operations so defined, for all P,Q ∈ Π(H): • P ′ = I − P , where I is the identity operator; • P ∨ Q is the projection onto the closed subspace generated by the union of the closed subspaces associated to P and Q. Although this model has triggered a conspicuous amount of research into orthomodular lattices (see e.g. [5, 34]), the relevance of such an approach for Presented by Heinrich Wansing; Received November 24, 2015 Studia Logica DOI: 10.1007/s11225-016-9670-3 c  Springer Science+Business Media Dordrecht 2016