Quasi‐splitting subspaces in a pre‐Hilbert space

Abstract

Key words Pre-Hilbert space (=inner product space), quasi-splitting subspace, orthomodular poset, lattice MSC (2000) 03G12, 46C05, 81P10 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.

Orthonormal bases and quasi-splitting subspaces in pre-Hilbert spaces

“…Motivated by the Amemiya-Araki-Piron Theorem, the authors of [4] conjectured that: E q (S) = E(S) if and only if S is a Hilbert space and also have settled this in the affirmative for the case when d(H/S) is finite. As will be seen further on, this question is closely related to the problem of characterizing those pre-Hilbert spaces that admit an orthonormal basis, i.e.…”

Affiliated subspaces and infiniteness of von Neumann algebras

“…M will be identified with B ( G )⊗ N , for some von Neumann algebra acting on K . By 3 there is a dense incomplete subspace S of G such that \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$E_M^q(S)\not= E_M(S)$\end{document}. Set Let X be a subspace of S that is quasi‐splitting but not splitting.…”

“…Recently, a few new directions have appeared in this research line. New type of a subspace, called quasi‐splitting subspace, was introduced and studied by Buhagiar, Chetcuti, Weber and others 3, 4. On the other hand, the investigation of subspace classes invariant with respect to a given von Neumann algebra has brought a new look at the link between the properties of subspace structures and structure theory of von Neumann algebra s. In particular, based on initial analysis in 10, 14, 15, the authors have shown in 8, that von Neumann algebra is properly infinite if, and only if, closed affiliated subspaces and operator closed affiliated subspaces coincide in the GNS representation space generated by a faithful normal state.…”

“…The symbols E M ( S ) and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$E^q_{M}(S)$\end{document} shall denote the set of all splitting and quasi‐splitting subspaces of S that are affiliated to M . The following hierarchy is obvious (except for may be the second inclusion proved in 3): …”

Probability structures in subspace lattice approach to foundations of quantum theory

“…We would like to summarize and refine our recent results in this area, bring new illustrating examples, and outline problems for future research. This contribution has grown out of the papers [3,4,5,6,9,10,11,12,14,15,16]. For basic treatment on this field, the reader can consult also monographs [7,8].…”