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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

46C99 Inner product spaces and their generalizations, Hilbert spaces
46A30 Open mapping and closed graph theorems; completeness (including \(B\)-, \(B_r\)-completeness)
Full Text: DOI
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