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David Buhagiar - One of the best experts on this subject based on the ideXlab platform.
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Order topology on orthocomplemented posets of linear subSpaces of a Pre-Hilbert Space
Annali di Matematica Pura ed Applicata (1923 -), 2020Co-Authors: David Buhagiar, E. Chetcuti, Hans WeberAbstract:Motivated by the Hilbert-Space model for quantum mechanics, we define a Pre-Hilbert Space logic to be a pair $$(S,{\mathscr {L}})$$ ( S , L ) , where S is a Pre-Hilbert Space and $${\mathscr {L}}$$ L is an orthocomplemented poset of orthogonally closed linear subSpaces of S , closed w.r.t. finite-dimensional perturbations (i.e., if $$M\in {\mathscr {L}}$$ M ∈ L and F is a finite-dimensional linear subSpace of S , then $$M+F\in {\mathscr {L}}$$ M + F ∈ L ). We study the order topology $$\tau _o({\mathscr {L}})$$ τ o ( L ) on $${\mathscr {L}}$$ L and show that completeness of S can by characterized by the separation properties of the topological Space $$({\mathscr {L}},\tau _o({\mathscr {L}}))$$ ( L , τ o ( L ) ) . It will be seen that the remarkable lack of a proper probability theory on Pre-Hilbert Space logics—for an incomplete S —comes out elementarily from this topological characterization.
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Order topology on orthocomplemented posets of linear subSpaces of a Pre-Hilbert Space
arXiv: Functional Analysis, 2018Co-Authors: David Buhagiar, Emmanuel Chetcuti, Hans WeberAbstract:Motivated by the Hilbert-Space model for quantum mechanics, we define a Pre-Hilbert Space logic to be a pair $(S,\el)$, where $S$ is a Pre-Hilbert Space and $\el$ is an orthocomplemented poset of orthogonally closed linear subSpaces of $S$, closed w.r.t. finite dimensional perturbations, (i.e. if $M\in\el$ and $F$ is a finite dimensional linear subSpace of $S$, then $M+F\in \el$). We study the order topology $\tau_o(\el)$ on $\el$ and show that completeness of $S$ can by characterized by the separation properties of the topological Space $(\el,\tau_o(\el))$. It will be seen that the remarkable lack of a proper probability-theory on Pre-Hilbert Space logics -- for an incomplete $S$ -- comes out elementarily from this topological characterization.
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Quasi-splitting subSpaces and Foulis-Randall subSpaces
Journal of Mathematical Physics, 2011Co-Authors: David Buhagiar, Emmanuel Chetcuti, Anatolij DvurečenskijAbstract:For a Pre-Hilbert Space S, let F(S) denote the orthogonally closed subSpaces, Eq(S) the quasi-splitting subSpaces, E(S) the splitting subSpaces, D(S) the Foulis-Randall subSpaces, and R(S) the maximal Foulis-Randall subSpaces, of S. It was an open problem whether the equalities D(S) = F(S) and E(S) = R(S) hold in general [Cattaneo, G. and Marino, G., “Spectral decomposition of Pre-Hilbert Spaces as regard to suitable classes of normal closed operators,” Boll. Unione Mat. Ital. 6 1-B, 451–466 (1982); Cattaneo, G., Franco, G., and Marino, G., “Ordering of families of subSpaces of Pre-Hilbert Spaces and Dacey Pre-Hilbert Spaces,” Boll. Unione Mat. Ital. 71-B, 167–183 (1987); Dvurecenskij, A., Gleason's Theorem and Its Applications (Kluwer, Dordrecht, 1992), p. 243.]. We prove that the first equality is true and exhibit a Pre-Hilbert Space S for which the second equality fails. In addition, we characterize complete Pre-Hilbert Spaces as follows: S is a Hilbert Space if, and only if, S has an orthonormal basis...
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Orthonormal bases and quasi-splitting subSpaces in Pre-Hilbert Spaces
Journal of Mathematical Analysis and Applications, 2008Co-Authors: David Buhagiar, Emmanuel Chetcuti, Hendrik WeberAbstract:Abstract Let S be a Pre-Hilbert Space. We study quasi-splitting subSpaces of S and compare the class of such subSpaces, denoted by E q ( S ) , with that of splitting subSpaces E ( S ) . In [D. Buhagiar, E. Chetcuti, Quasi splitting subSpaces in a Pre-Hilbert Space, Math. Nachr. 280 (5–6) (2007) 479–484] it is proved that if S has a non-zero finite codimension in its completion, then E q ( S ) ≠ E ( S ) . In the present paper it is shown that if S has a total orthonormal system, then E q ( S ) = E ( S ) implies completeness of S . In view of this result, it is natural to study the problem of the existence of a total orthonormal system in a Pre-Hilbert Space. In particular, it is proved that if every algebraic complement of S in its completion is separable, then S has a total orthonormal system.
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Quasi-splitting subSpaces in a Pre-Hilbert Space
Mathematische Nachrichten, 2007Co-Authors: David Buhagiar, Emmanuel ChetcutiAbstract:Let S be a Pre-Hilbert Space. Two classes of closed subSpaces of S that can naturally replace the lattice of projections in a Hilbert Space are E (S) and F (S), the classes of splitting subSpaces and orthogonally closed subSpaces of S respectively. It is well-known that in general the algebraic structure of E (S) differs considerably from that of F (S) and the two coalesce if and only if S is a Hilbert Space. In the present note we introduce the class Eq(S) of quasi-splitting subSpaces of S. First it is shown that Eq(S) falls between E (S) and F (S). It is also shown that, in contrast to the other two classes, Eq(S) can sometimes be a complete lattice (without S being complete) and yet, in other examples Eq(S) is not a lattice. At the end, the algebraic structure of Eq(S) is used to characterize Hilbert Spaces. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
Emmanuel Chetcuti - One of the best experts on this subject based on the ideXlab platform.
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Order topology on orthocomplemented posets of linear subSpaces of a Pre-Hilbert Space
arXiv: Functional Analysis, 2018Co-Authors: David Buhagiar, Emmanuel Chetcuti, Hans WeberAbstract:Motivated by the Hilbert-Space model for quantum mechanics, we define a Pre-Hilbert Space logic to be a pair $(S,\el)$, where $S$ is a Pre-Hilbert Space and $\el$ is an orthocomplemented poset of orthogonally closed linear subSpaces of $S$, closed w.r.t. finite dimensional perturbations, (i.e. if $M\in\el$ and $F$ is a finite dimensional linear subSpace of $S$, then $M+F\in \el$). We study the order topology $\tau_o(\el)$ on $\el$ and show that completeness of $S$ can by characterized by the separation properties of the topological Space $(\el,\tau_o(\el))$. It will be seen that the remarkable lack of a proper probability-theory on Pre-Hilbert Space logics -- for an incomplete $S$ -- comes out elementarily from this topological characterization.
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Quasi-splitting subSpaces and Foulis-Randall subSpaces
Journal of Mathematical Physics, 2011Co-Authors: David Buhagiar, Emmanuel Chetcuti, Anatolij DvurečenskijAbstract:For a Pre-Hilbert Space S, let F(S) denote the orthogonally closed subSpaces, Eq(S) the quasi-splitting subSpaces, E(S) the splitting subSpaces, D(S) the Foulis-Randall subSpaces, and R(S) the maximal Foulis-Randall subSpaces, of S. It was an open problem whether the equalities D(S) = F(S) and E(S) = R(S) hold in general [Cattaneo, G. and Marino, G., “Spectral decomposition of Pre-Hilbert Spaces as regard to suitable classes of normal closed operators,” Boll. Unione Mat. Ital. 6 1-B, 451–466 (1982); Cattaneo, G., Franco, G., and Marino, G., “Ordering of families of subSpaces of Pre-Hilbert Spaces and Dacey Pre-Hilbert Spaces,” Boll. Unione Mat. Ital. 71-B, 167–183 (1987); Dvurecenskij, A., Gleason's Theorem and Its Applications (Kluwer, Dordrecht, 1992), p. 243.]. We prove that the first equality is true and exhibit a Pre-Hilbert Space S for which the second equality fails. In addition, we characterize complete Pre-Hilbert Spaces as follows: S is a Hilbert Space if, and only if, S has an orthonormal basis...
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Orthonormal bases and quasi-splitting subSpaces in Pre-Hilbert Spaces
Journal of Mathematical Analysis and Applications, 2008Co-Authors: David Buhagiar, Emmanuel Chetcuti, Hendrik WeberAbstract:Abstract Let S be a Pre-Hilbert Space. We study quasi-splitting subSpaces of S and compare the class of such subSpaces, denoted by E q ( S ) , with that of splitting subSpaces E ( S ) . In [D. Buhagiar, E. Chetcuti, Quasi splitting subSpaces in a Pre-Hilbert Space, Math. Nachr. 280 (5–6) (2007) 479–484] it is proved that if S has a non-zero finite codimension in its completion, then E q ( S ) ≠ E ( S ) . In the present paper it is shown that if S has a total orthonormal system, then E q ( S ) = E ( S ) implies completeness of S . In view of this result, it is natural to study the problem of the existence of a total orthonormal system in a Pre-Hilbert Space. In particular, it is proved that if every algebraic complement of S in its completion is separable, then S has a total orthonormal system.
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Quasi-splitting subSpaces in a Pre-Hilbert Space
Mathematische Nachrichten, 2007Co-Authors: David Buhagiar, Emmanuel ChetcutiAbstract:Let S be a Pre-Hilbert Space. Two classes of closed subSpaces of S that can naturally replace the lattice of projections in a Hilbert Space are E (S) and F (S), the classes of splitting subSpaces and orthogonally closed subSpaces of S respectively. It is well-known that in general the algebraic structure of E (S) differs considerably from that of F (S) and the two coalesce if and only if S is a Hilbert Space. In the present note we introduce the class Eq(S) of quasi-splitting subSpaces of S. First it is shown that Eq(S) falls between E (S) and F (S). It is also shown that, in contrast to the other two classes, Eq(S) can sometimes be a complete lattice (without S being complete) and yet, in other examples Eq(S) is not a lattice. At the end, the algebraic structure of Eq(S) is used to characterize Hilbert Spaces. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
Walter Benz - One of the best experts on this subject based on the ideXlab platform.
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Lie Sphere Geometry in Hubert Spaces
Results in Mathematics, 2001Co-Authors: Walter BenzAbstract:We develop Lie sphere geometry for arbitrary real Pre-Hilbert Spaces of (finite or infinite) dimension at least 2. One of the main results is that a bijection of the set of all Laguerre cycles which preserves contact in one direction must already be a Lie transformation (THEOREM 2). As a first consequence of this theorem we get that a bijection of an arbitrary real Pre-Hilbert Space of dimension at least 3 which preserves Lorentz-Minkowski distance 0 in one direction must already be a (proper or improper) Lorentz boost up to a dilatation, a translation and an orthogonal mapping (THEOREM 3). This is a generalization of results of A.D. Alexandroff [ 1 ], E.M. Schröder [ 21 ] and F. Cacciafesta [ 7 ]. Another consequence is that a bijection of the set of all Lie cycles which preserves contact in one direction must already be a Lie transformation (THEOREM 4). If we apply this result to the finite dimensional case, we get that the diffeomorphism assumption in the Fundamental Theorem of Lie sphere geometry as stated in Theorem 1.5 in T.E. Cecil [ 8 ], p. 33, is not needed for the proof of this theorem (REMARK to THEOREM 4).
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Elliptic distances in Hilbert Spaces
Aequationes Mathematicae, 2000Co-Authors: Walter BenzAbstract:Let X be a Pre-Hilbert Space of dimension at least 2 and define \( X_0 := X\backslash\{0\} \). In Theorems 1 and 2 we determine all \( f : X_0 \to X_0 \) satisfying the functional equation¶¶\( \forall_{x,y\in X_0}\quad {|f(x)f(y)|\over||f(x)||\cdot||f(y)||} = {|xy|\over||x|| \cdot ||y||} \).,¶and also all \( f : X_0\to X_0 \) with¶¶\( \forall_{x,y\in X_0}\quad {|f(x)f(y)|\over||f(x)||\cdot||f(y)||} = {|xy|\over||x|| \cdot ||y||} \).¶In Theorem 6 we characterize the functions¶¶\( \varepsilon: X_0 \times X_0\to\bigl[0,{\pi\over2}\bigr] \) and \( \sigma : X_0 \times X_0\to [0,\pi] \)¶satisfying¶\( ||x|| \cdot ||y|| \cdot \cos \varepsilon \,(x,y) = |xy| \),¶\( ||x|| \cdot ||y|| \cdot \cos \sigma \,(x,y) = xy \),¶respectively.
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Hyperbolic distances in Hilbert Spaces
Aequationes mathematicae, 1999Co-Authors: Walter BenzAbstract:We present a functional equations approach to the non-negative functions $ E \,(x,y) $ and $ E\,(x,y) $ satisfying¶¶ $ {\rm cosh}\,h (x,y) = \sqrt{1+x^2}\,\sqrt{1+y^2} - xy $ ,¶ $ E\,(x,y) = ||x - y|| $ .¶ The underlying structure is a Pre-Hilbert Space X of dimension at least 2. An important tool is the group of translations¶¶ $ T_t(x) = x + \bigl((xe)({\rm cosh}\,t - 1) + \sqrt{1+x^2}\,{\rm sinh} \,t\bigr)\,e $ ,¶ $ t \in {\Bbb R} $ , where $ T_t:X \to X $ satisfies the translation equation with a fixed $ e\in X $ such that $ e^2 = 1 $ . One of the results is that a function¶¶ $ d : X \times X \to {\Bbb R}_{\geq 0} : = {r \in {\Bbb R} \mid\,r \geq 0} $ ¶which is invariant under orthogonal mappings and the described translations for a fixed e , must be of the form¶¶ $ d\,(x,y) = g\,\bigl((h(x,y)\bigr) $ ¶with an arbitrary function $ g:{\Bbb R}_{\geq 0}\to{\Bbb R}_{\geq 0} $ . If, moreover, d is additive on the line $ {\xi e \mid \xi \in{\Bbb R}} $ , then d is essentially equal to h .
Hans Weber - One of the best experts on this subject based on the ideXlab platform.
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Order topology on orthocomplemented posets of linear subSpaces of a Pre-Hilbert Space
Annali di Matematica Pura ed Applicata (1923 -), 2020Co-Authors: David Buhagiar, E. Chetcuti, Hans WeberAbstract:Motivated by the Hilbert-Space model for quantum mechanics, we define a Pre-Hilbert Space logic to be a pair $$(S,{\mathscr {L}})$$ ( S , L ) , where S is a Pre-Hilbert Space and $${\mathscr {L}}$$ L is an orthocomplemented poset of orthogonally closed linear subSpaces of S , closed w.r.t. finite-dimensional perturbations (i.e., if $$M\in {\mathscr {L}}$$ M ∈ L and F is a finite-dimensional linear subSpace of S , then $$M+F\in {\mathscr {L}}$$ M + F ∈ L ). We study the order topology $$\tau _o({\mathscr {L}})$$ τ o ( L ) on $${\mathscr {L}}$$ L and show that completeness of S can by characterized by the separation properties of the topological Space $$({\mathscr {L}},\tau _o({\mathscr {L}}))$$ ( L , τ o ( L ) ) . It will be seen that the remarkable lack of a proper probability theory on Pre-Hilbert Space logics—for an incomplete S —comes out elementarily from this topological characterization.
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Order topology on orthocomplemented posets of linear subSpaces of a Pre-Hilbert Space
arXiv: Functional Analysis, 2018Co-Authors: David Buhagiar, Emmanuel Chetcuti, Hans WeberAbstract:Motivated by the Hilbert-Space model for quantum mechanics, we define a Pre-Hilbert Space logic to be a pair $(S,\el)$, where $S$ is a Pre-Hilbert Space and $\el$ is an orthocomplemented poset of orthogonally closed linear subSpaces of $S$, closed w.r.t. finite dimensional perturbations, (i.e. if $M\in\el$ and $F$ is a finite dimensional linear subSpace of $S$, then $M+F\in \el$). We study the order topology $\tau_o(\el)$ on $\el$ and show that completeness of $S$ can by characterized by the separation properties of the topological Space $(\el,\tau_o(\el))$. It will be seen that the remarkable lack of a proper probability-theory on Pre-Hilbert Space logics -- for an incomplete $S$ -- comes out elementarily from this topological characterization.
Joseph A. Wolf - One of the best experts on this subject based on the ideXlab platform.
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Infinite dimensional multiplicity free Spaces III: matrix coefficients and regular functions
Mathematische Annalen, 2011Co-Authors: Joseph A. WolfAbstract:In earlier papers we studied direct limits $${(G,\,K) = \varinjlim\, (G_n,K_n)}$$ of two types of Gelfand pairs. The first type was that in which the G _ n / K _ n are compact Riemannian symmetric Spaces. The second type was that in which $${G_n = N_n\rtimes K_n}$$ with N _ n nilpotent, in other words pairs ( G _ n , K _ n ) for which G _ n / K _ n is a commutative nilmanifold. In each we worked out a method inspired by the Frobenius–Schur Orthogonality Relations to define isometric injections $${\zeta_{m,n}: L^2(G_n/K_n) \hookrightarrow L^2(G_m/K_m)}$$ for m ≧ n and prove that the left regular representation of G on the Hilbert Space direct limit $${L^2(G/K) := \varinjlim L^2(G_n/K_n)}$$ is multiplicity-free. This left open questions concerning the nature of the elements of L ^2( G / K ). Here we define Spaces $${\mathcal{A}(G_n/K_n)}$$ of regular functions on G _ n / K _ n and injections $${\nu_{m,n} : \mathcal{A}(G_n/K_n) \to \mathcal{A}(G_m/K_m)}$$ for m ≧ n related to restriction by $${\nu_{m,n}(f)|_{G_n/K_n} = f}$$ . Thus the direct limit $${\mathcal{A}(G/K) := \varinjlim \{\mathcal{A}(G_n/K_n), \nu_{m,n}\}}$$ sits as a particular G -submodule of the much larger inverse limit $${\varprojlim \{\mathcal{A}(G_n/K_n), {\rm restriction}\}}$$ . Further, we define a pre Hilbert Space structure on $${\mathcal{A}(G/K)}$$ derived from that of L ^2( G / K ). This allows an interpretation of L ^2( G / K ) as the Hilbert Space completion of the concretely defined function Space $${\mathcal{A}(G/K)}$$ , and also defines a G -invariant inner product on $${\mathcal{A}(G/K)}$$ for which the left regular representation of G is multiplicity-free.
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Infinite dimensional multiplicity free Spaces III: matrix coefficients and regular functions
Mathematische Annalen, 2010Co-Authors: Joseph A. WolfAbstract:In earlier papers we studied direct limits (G, K ) = lim → (Gn, Kn) of two types of Gelfand pairs. The first type was that in which the Gn/Kn are compact Riemannian symmetric Spaces. The second type was that in which Gn = NnKn with Nn nilpotent, in other words pairs (Gn, Kn) for which Gn/Kn is a commutative nilmanifold.IneachweworkedoutamethodinspiredbytheFrobenius-SchurOrthog- onality Relations to define isometric injections ζm,n : L 2 (Gn/Kn )� → L 2 (Gm/Km) for m n and prove that the left regular representation of G on the Hilbert Space directlimit L 2 (G/K ) := lim → L 2 (Gn/Kn)ismultiplicity-free.Thisleftopenquestions concerning the nature of the elements of L 2 (G/K ). Here we define SpacesA(Gn/Kn) of regular functions on Gn/Kn and injections νm,n : A(Gn/Kn) → A(Gm/Km) for m n related to restriction by νm,n( f )|Gn /Kn = f . Thus the direct limit A(G/K ) := lim → {A(Gn/Kn), νm,n} sits as a particular G-submodule of the much larger inverse limit lim − {A(Gn/Kn),restriction}. Further, we define a pre Hilbert Space structure on A(G/K ) derived from that of L 2 (G/K ). This allows an interpretation of L 2 (G/K ) as the Hilbert Space completion of the concretely defined function Space A(G/K ), and also defines a G-invariant inner product on A(G/K ) for which the left regular representation of G is multiplicity-free.
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Infinite Dimensional Multiplicity Free Spaces III: Matrix Coefficients and Regular Functions
arXiv: Representation Theory, 2009Co-Authors: Joseph A. WolfAbstract:In earlier papers we studied direct limits $(G,K) = \varinjlim (G_n,K_n)$ of two types of Gelfand pairs. The first type was that in which the $G_n/K_n$ are compact Riemannian symmetric Spaces. The second type was that in which $G_n = N_n\rtimes K_n$ with $N_n$ nilpotent, in other words pairs $(G_n,K_n)$ for which $G_n/K_n$ is a commutative nilmanifold. In each we worked out a method inspired by the Frobenius--Schur Orthogonality Relations to define isometric injections $\zeta_{m,n}: L^2(G_n/K_n) \hookrightarrow L^2(G_m/K_m)$ for $m \geqq n$ and prove that the left regular representation of $G$ on the Hilbert Space direct limit $L^2(G/K) := \varinjlim L^2(G_n/K_n)$ is multiplicity--free. This left open questions concerning the nature of the elements of $L^2(G/K)$. Here we define Spaces $\cA(G_n/K_n)$ of regular functions on $G_n/K_n$ and injections $\nu_{m,n} : \cA(G_n/K_n) \to \cA(G_m/K_m)$ for $m \geqq n$ related to restriction by $\nu_{m,n}(f)|_{G_n/K_n} = f$. Thus the direct limit $\cA(G/K):= \varinjlim \{\cA(G_n/K_n), \nu_{m,n}\}$ sits as a particular $G$--submodule of the much larger inverse limit $\varprojlim \{\cA(G_n/K_n), \text{restriction}\}$. Further, we define a pre Hilbert Space structure on $\cA(G/K)$ derived from that of $L^2(G/K)$. This allows an interpretation of $L^2(G/K)$ as the Hilbert Space completion of the concretely defined function Space $\cA(G/K)$, and also defines a $G$--invariant inner product on $\cA(G/K)$ for which the left regular representation of $G$ is multiplicity--free.
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Infinite Dimensional Multiplicity Free Spaces III
2009Co-Authors: Joseph A. WolfAbstract:In earlier papers we studied direct limits (G, K) = lim → (Gn, Kn) of two types of Gelfand pairs. The first type was that in which the Gn/Kn are compact riemannian symmetric Spaces. The second type was that in which Gn = Nn ⋊ Kn with Nn nilpotent, in other words pairs (Gn, Kn) for which Gn/Kn is a commutative nilmanifold. In each we worked out a method inspired by the Frobenius–Schur Orthogonality Relations to define isometric injections ζm,n : L 2 (Gn/Kn) → L 2 (Gm/Km) for m ≧ n and prove that the left regular representation of G on the Hilbert Space direct limit L 2 (G/K) := lim → L 2 (Gn/Kn) is multiplicity–free. This left open questions concerning the nature of the elements of L 2 (G/K). Here we define Spaces A(Gn/Kn) of regular functions on Gn/Kn and injections νm,n : A(Gn/Kn) → A(Gm/Km) for m ≧ n related to restriction by νm,n(f)|Gn/Kn = f. Thus the direct limit A(G/K) := lim → {A(Gn/Kn), νm,n} sits as a particular G–submodule of the much larger inverse limit lim − {A(Gn/Kn),restriction}. Further, we define a pre Hilbert Space structure on A(G/K) derived from that of L 2 (G/K). This allows an interpretation of L 2 (G/K) as the Hilbert Space completion of the concretely defined function Space A(G/K), and also defines a G–invariant inner product on A(G/K) for which the left regular representation of G is multiplicity–free.