Decoherence in Pre-symmetric Spaces

Résumé

Pre-symmetric complex Banach spaces have been proposed as models for state spaces of physical systems. A structural projection on a pre-symmetric space A∗ represents an operation on the corresponding system, and has as its range a further pre-symmetric space which represents the state space of the resulting system and symmetries of the system are represented by elements of the group Aut(A∗) of linear isometries of A∗. Two structural projections R and S on the pre-symmetric space A∗ represent decoherent operations when their ranges are rigidly collinear. It is shown that, for decoherent elements x and y of A∗, there exists an involutive element φ∗ in Aut(A∗) which conjugates the structural projections corresponding to x and y, and conditions are found for φ∗ to exchange x and y. The results are used to investigate when certain subspaces of A∗ are the ranges of contractive projections and, therefore, represent systems arising from filtering operations.