Jordan decomposition of measures on projections in J-spaces (Q804912)

Let H be a space with an indefinite inner product [,] and the canonical symmetry J (i.e. H is a J-space). H is a Hilbert space with respct to the inner product \((x,y)=[Jx,y]\). Denote by \({\mathcal P}\) (resp. \({\mathcal P}^+,{\mathcal P}^-)\) the set of all J-self-adjoint bounded projections p (resp. with positive, resp. negative subspace pH). A map \(\nu\) :\({\mathcal P}\to {\mathbb{R}}\) is a measure if \(\nu (\Sigma p_ i)=\Sigma \nu (p_ i)\) for any orthogonal family \(\{p_ i\}\subset {\mathcal P}\). The measure \(\nu\) is called bounded if \(\sup \{| \nu (p)| /\| p\|,p\in {\mathcal P}\}<\infty\). A measure \(\nu\) is said to be indefinite if \(\nu| {\mathcal P}^+\geq 0,\nu |_{{\mathcal P}}-\leq 0\). In the previous paper the author obtained an analogue of Gleason theorem for bounded measures on J- spaces. In this paper he gives the following Corollary 1. (Jordan decomposition) Any bounded measure in a J-space of dimension at least three, is a linear combination of indefinite measures. The author proves also the following Theorem 2. In infinite dimensional J-space any measure \(\nu\vdots {\mathcal P}\to {\mathbb{R}}\) is bounded.