# zbMATH — the first resource for mathematics

State on splitting subspaces and completeness of inner product spaces. (English) Zbl 0661.46019
Let V be a real or complex inner product space, $${\mathcal L}(V)=\{M\subseteq V:M^{\perp \perp}=M\}$$, and $${\mathcal E}(V)=\{M\subseteq V:M+M^{\perp}=V\}$$. The element of $${\mathcal E}(V)$$ is called a splitting subspace, and $${\mathcal L}(V)\supseteq {\mathcal E}(V)$$ holds. Let m be a mapping from $${\mathcal E}(V)$$ into the interval [0,1] satisfying the following conditions (i) and (ii): (i) $$m(V)=1$$, (ii) Let $$\{M_ t\in {\mathcal E}(V):t\in T\}$$ be a system of mutually orthogonal subspaces. If $$\bigvee_{L,t\in T}M_ t\equiv span(\cup_{t\in T}M_ t)^{\perp \perp}\in {\mathcal E}(V)$$, then m($$\bigvee_{L,t\in T}M_ t)=\sum_{t\in T}m(M_ t)$$ holds.
If the condition (ii) holds for any index set T, m is said to be a completely additive state (C.A. state). The authors prove the following main theorem here: An inner product space V is complete if and only if $${\mathcal E}(V)$$ possesses at least one C.A. state.
Let T be a positive Hermitian operator from V into V with trace equal to 1 and $$P_ M$$ the orthoprojector from V onto M. The necessity, that is the existence of C. A. state, is easily proved by the construction of $$m_ T(M)=tr(TP_ M)$$. They give the following two different proofs of the sufficiency: In Proof 1 they show that, for any sequence of orthonormal vectors (O.N. vectors) $$\{x_ i\}^{\infty}_{i=1}$$ from V, the space $$M=\bigvee^{\infty}_{i=1,L}sp(x_ i)$$ is complete and splitting. Then V is complete from the criterion by A. Dvurečeskij [Lett. Math. Phys. 15, 231-235 (1988)]. In Proof 2 they show that, for any nonvoid system of O.N. vectors from V, $$A=\bigvee_{i,L}sp(x_ i)$$ is splitting. Then $${\mathcal E}(V)\supseteq {\mathcal L}(V)$$ is proved by using a maximal orthonormal system $$\{y_ i\}$$ in M, and V is complete. The C. A. state is used in fundamental Lemmas.
Reviewer: H.Yamagata

##### MSC:
 46C99 Inner product spaces and their generalizations, Hilbert spaces 46A30 Open mapping and closed graph theorems; completeness (including $$B$$-, $$B_r$$-completeness)
Full Text:
##### References:
 [1] Amemiya, I., and Araki, H. (1966). A remark on Piron’s paper,Publication of the Research Institute for Mathematical Sciences, Series A,2, 423–427. · Zbl 0177.16103 [2] Cattaneo, G., and Marino, G. (1986). Completeness of inner product spaces with respect to splitting subspaces.Letters in Mathematical Physics,11, 15–20. · Zbl 0601.46026 [3] Dvurečenskij, A. (1988a). Note on a construction of unbounded measures on a nonseparable Hubert space quantum logic.Annales de l’Institut Henri Poincaré-Physique Theorique,48, 297–310. [4] Dvurečenskij, A. (1988b). Completeness of inner product spaces and quantum logic of splitting subspaces.Letters in Mathematical Physics,15, 231–235. · Zbl 0652.46017 [5] Dvurečenskij, A. (1989). A state criterion of the completeness for inner product spaces,Demonstratio Mathematique, in press. · Zbl 0722.46010 [6] Dvurečenskij, A., and Misik, Jr., L. (1988). Gleason’s theorem and completeness of inner product spaces,International Journal of Theoretical Physics,27, 417–426. · Zbl 0663.46060 [7] Gross, H., and Keller, H. A. (1977). On the definition of Hubert space.Manuscripta Mathematica,23, 67–90. · Zbl 0365.46023 [8] Gudder, S. P. (1974). Inner product spaces,American Mathematical Monthly,81, 29–36. · Zbl 0279.46013 [9] Hamhalter, J., and Pták, P. (1987). A completeness criterion for inner product spaces,19, 259–263. · Zbl 0601.46027 [10] Maeda, S. (1980).Lattice Theory and Quantum Logic, Mahishoten, Tokyo (in Japanese). · Zbl 0446.70022 [11] Varadarajan, V. S. (1962). Probability in physics and a theorm on simultaneous observability,Communications in Pure and Applied Mathematics,15, 186–217 [Errata,18 (1965)]. · Zbl 0109.44705 [12] Varadarajan, V. S. (1968).Geometry of Quantum Theory, Vol. 1, Van Nostrand, Princeton, New Jersey. · Zbl 0155.56802
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.