Completeness Of Inner Product Spaces Research Articles

Completeness of Inner Product Spaces Associated with Functional on Jordan Structures

We show that a normal functional $$\varphi$$ on a $$JBW^{\ast}$$ triple induces, via Gelfand–Naimark–Segal like construction, a complete inner product space if and only if $$\varphi$$ is a finite convex combination of extreme points from the predual. Application of this result to von Neumann algebras is shown.

Completeness of Inner Product Spaces Induced by States on Jordan and C ∗-Algebras

We show that a state φ on a Jordan algebra 𝓐 induces complete inner product space if and only if φ is a convex combination of pure states. Inner product spaces generated by Type I n factor states and states on spin factors are described. We initiate study of completely positive maps in this connection by showing that pure completely positive map on a C ∗-algebra gives always complete inner product space in the Stinespring construction.

Classes of Invariant Subspaces for Some Operator Algebras

New results showing connections between structural properties of von Neumann algebras and order theoretic properties of structures of invariant subspaces given by them are proved. We show that for any properly infinite von Neumann algebra M there is an affiliated subspace \({\mathcal{L}} \) such that all important subspace classes living on \({\mathcal{L}} \) are different. Moreover, we show that \({\mathcal{L}} \) can be chosen such that the set of σ-additive measures on subspace classes of \({\mathcal{L}} \) are empty. We generalize measure theoretic criterion on completeness of inner product spaces to affiliated subspaces corresponding to Type I factor with finite dimensional commutant. We summarize hitherto known results in this area, discuss their importance for mathematical foundations of quantum theory, and outline perspectives of further research.

Completeness of Inner Product Spaces and G. Cattaneo

In the Eighties G. Cattaneo contributed to Hilbert space quantum mechanics models using the family of splitting subspaces of an inner product space in order to find completeness criteria. In this p...

Completeness of Inner Product Spaces and G. Cattaneo

In the Eighties G. Cattaneo contributed to Hilbert space quantum mechanics models using the family of splitting subspaces of an inner product space in order to find completeness criteria. In this paper we show how that theory was developed in the last 25 years into a rich theory which has a close connection to quantum information. G. Cattaneo also posed an open problem which was solved only this year.

Boundedness of sign-preserving charges, regularity, and the completeness of inner product spaces

AbstractWe introduce sign-preserving charges on the system of all orthogonally closed subspaces, F(S), of an inner product space S, and we show that it is always bounded on all the finite-dimensional subspaces whenever dim S = ∞. When S is finite-dimensional this is not true. This fact is used for a new completeness criterion showing that S is complete whenever F(S) admits at least one non-zero sign-preserving regular charge. In particular, every such charge is always completely additive.

A Finitely Additive State Criterion for the Completeness of Inner Product Spaces

We show that an inner product space S (real, complex or quaternion) is complete if, and only if, the system of all orthogonally closed subspaces in S, denoted by F(S), admits at least one finitely additive state which is not vanishing on the set of all finite dimensional subspaces of S. Although it gives only a partial solution to the problem formulated by Ptak on the existence of a finitely additive state on F(S) for incomplete S, this gives an important insight into the structure of the set of states on F(S). This criterion has no analogue whatsoever in E(S), the system of splitting subspaces of S.

A New Algebraic Criterion for Completeness of Inner Product Spaces

We show that an inner product space S is complete whenever the system E(S) of all splitting subspaces of S, i.e., of all subspaces M of S such that M + M ⊥ = S holds, satisfies the σ-Riesz interpolation property. This generalizes the result of H. Gross and H. Keller who required E(S) to be a complete lattice, of G. Cattaneo and G. Marino who required E(S) to be a σ-complete lattice, and that of the author who required E(S) to be a σ-orthocomplete OMP.

Test spaces, dacey spaces, and completeness of inner product spaces

We show that a test space consisting of the unit sphere of a real or complex inner product spaceS and of the set all maximal orthonormal systems inS, is algebraic iffS is Dacey or, equivalently, iffS is complete. In addition, we present another completeness criterion.

Finitely additive states and completeness of inner product spaces

For any unit vector in an inner product space S, we define a mapping on the system of all ⊥-closed subspaces of S, F(S), whose restriction on the system of all splitting subspaces of S, E(S), is always a finitely additive state. We show that S is complete iff at least one such mapping is a finitely additive state on F(S). Moreover, we give a completeness criterion via the existence of a regular finitely additive state on appropriate systems of subspaces. Finally, the result will be generalized to general inner product spaces.

State on splitting subspaces and completeness of inner product spaces

We show that an inner produce space V is complete if the system of all splitting subspaces, i.e., of all subspaces M for which M + M/sup perpenficular/ = V, possesses at least one completely additive state.

Completeness of inner product spaces and quantum logic of splitting subspaces

We show that an inner product space S is complete whenever its system E(S) of all splitting subspaces, i.e., of all subspaces E for which E⊕E⊥=S holds, is a quantum logic, that is, an orthocomplemented orthomodular σ-orthoposet. It is well known that the quantum logic is an important axiomatic model of quantum mechanics. This generalizes the result of G. Cattaneo and G. Marino (Lett. Math. Phys.11, 15–20 (1986)) who required that E(S) be a lattice. Moreover, the conditions are weakened to show that S is complete whenever E(S) contains the join of any sequence of one-dimentional orthogonal subspaces.

Gleason's theorem and completeness of inner product spaces

We generalize the result of Hamhalter and Ptak showing that an inner product space whose dimension is either a nonmeasurable cardinal or an arbitrary cardinal is complete iff its lattice of strongly closed subspaces possesses at least one state or one completely additive state, respectively. Moreover, we show that this lattice of any separable space possesses manyσ-finite measures, and we give the Gleason analogue for them.

Completeness of inner product spaces with respect to splitting subspaces

The set E(S) of all splitting subspaces, i.e., of all subspaces M of an inner product space S for which M⊕M ⊥=S holds, is an orthocomplemented orthomodular orthoposet and it has already been shown that the ordering property on E(S) of being a complete lattice characterizes the completeness of inner product spaces. In this work this last result is generalized proving that S is a Hilbert space under the weaker request that E(S) is a σ-lattice. As a marginal result, we also prove that an inner product space is complete if and only if the complete lattice ∑(S) of all subspaces is orthomodular.