# Article

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Keywords:
Hilbert-space-valued state; $h$-state; finitely additive orthogonal vector states; normed measures; Hilbert logic; extension property
Summary:
We analyze finitely additive orthogonal states whose values lie in a real Hilbert space. We call them $h$-states. We first consider the important case of $h$-states on a standard Hilbert logic $L(H)$ of projectors in $H$-we describe the $h$-states $s$: $L(H_1) \rightarrow H_2$, where $\text {dim } H_2 \leq$ \text {dim} H_1 < \infty$. In particular, we show that, up to a unitary mapping, every$h$-state$s$:$L(H)\rightarrow H(3\leq \text {dim } H < \infty)$has to be concentrated on a one-dimensional projection. We also study the$h$-states$s$:$L(H_1)\rightarrow H_2$for the case of$\text {dim } H_1 = \infty$. The results of the first part complement the papers [10] and [13]. In the second part we investigate$h$-states on general logics. Being motivated by the quantum axiomatics, the main question we ask here is as follow: Given a Hilbert space$H$with$\text {dim } H < \infty$, what is the class of such logics$L$that, for any Boolean subalgebra$B$of$L$, every$h$-states$s$:$B \rightarrow H$extends over$L\$? We answer this question by finding a simple condition characterizing the class (Theorem 3.4]. It turns out that the class is considerably large-it contains e.g. all concrete logics-but, on the other hand, it does not contain all finite logics (we construct a counterexample in the appendix).
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