Decoherence in pre-symmetric spaces

Pre-symmetric complex Banach spaces have been proposed as models for state spaces of physical systems. A structural projection on a pre-symmetric space At 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 axe 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.