Binary Structures on Banach Spaces

The aim of the present work is to give a mathematical underpinning for the use of quasi-probabilities and pseudo-metrics in infinite-dimensional Banach manifolds. The notion of a continuous binary structure is introduced. It is a triple consisting of a continuous symmetric bilinear form together with a pair of closed linear subspaces of a Banach space. Such binary structures are abundant in Hilbert spaces. In order to confirm their existence in arbitrary Banach spaces, the auxiliary notion is introduced of subspaces that are positive with respect to a given symmetric bilinear form. It is shown that any subspace which is maximally positive with respect to the bilinear form induces a continuous binary structure on the Banach space. The Wigner function of a system of quantum mechanical particles is treated as an example.