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

A geometric characterization of structural projections on a JBW*-triple

AbstractA structural projection R on a Jordan∗-triple A is a linear projection such that, for all elements a, b and c in A,R{aRbc}={RabRc}.The L-orthogonal complement G◊ of a subset G of a complex Banach space E is the set of elements x in E such that, for all elements y in G,||x±y||=||x||+||y||.A contractive projection P on E is said to be neutral if the condition that||Px||=||x||implies that the elements Px and x coincide, and is said to be a GL-projection if the L-orthogonal complement (PE)◊ of the range PE of P is contained in the kernel ker(P) of P. It is shown that, for a JBW∗-triple A, with predual A∗, a linear projection R on A is structural if and only if it is the adjoint of a neutral GL-projection on A∗, thereby giving a purely geometric characterization of structural projections