Algebraic properties of complex fuzzy events in classical and in quantum information systems (Q1090664)

The algebraic properties of fuzzy events, both classical and quantum, are described by using the language of Baer*-semigroups. It has been shown that the set of all classical fuzzy events is a commutative Baer*- semigroup with trivial involution and Brouwerian complementation. The family of all complex fuzzy events is shown to be a commutative Baer*- semigroup with nontrivial involution and Brouwerian complementation, of which the Baer*-semigroup of (classical) fuzzy events is a sub-Baer*- semigroup. The set of all quantum fuzzy events (i.e., bounded selfadjoint operators F on a Hilbert space such that \(0\leq F\leq I)\) is shown to be a partial Baer*-semigroup (in which the multiplication is defined for couples of commutative elements only) with trivial involution and generalized Brouwerian complementation. The collection of all complex quantum fuzzy events is a noncommutative Baer*-semigroup with nontrivial involution and generalized Brouwerian complementation. Finally, it is shown that both exact and fuzzy quantum events can be represented as classical fuzzy events.