Kadison’s antilattice theorem for a synaptic algebra

Foulis, David J.; Pulmannová, Sylvia

doi:10.1515/dema-2018-0002

Cited by 2 publications

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“…Our purpose in this paper, as indicated by its title, is to study the spectral order on a so-called synaptic algebra (abbreviated SA) [7,12,13,14,15,16,17,18,19,20,21,22,40]. The formulation of a synaptic algebra [7] was motivated by an effort to define, by means of a few simple and physically plausible axioms, an algebraic structure suitable for the mathematical description of a quantum mechanical system.…”

Section: Introductionmentioning

confidence: 99%

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Spectral Order on a Synaptic Algebra

Foulis

Pulmannová

2018

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We define and study an alternative partial order, called the spectral order, on a synaptic algebra-a generalization of the self-adjoint part of a von Neumann algebra. We prove that if the synaptic algebra A is norm complete (a Banach synaptic algebra), then under the spectral order, A is Dedekind σ-complete lattice, and the corresponding effect algebra E is a σ-complete lattice. Moreover, E can be organized into a Brouwer-Zadeh algebra in both the usual (synaptic) and spectral ordering; and if A is Banach, then E is a Brouwer-Zadeh lattice in the spectral ordering. If A is of finite type, then De Morgan laws hold on E in both the synaptic and spectral ordering.An existing infimum (respectively, supremum) of a family (a γ ) γ∈Γ in P is written as γ∈Γ a γ (respectively, as γ∈Γ a γ ).We denote the von Neumann algebra of all bounded linear operators on a Hilbert space H by B(H) and we denote the self-adjoint part of B(H) by B sa (H). If ·, · is the inner product on H, then the usual order ≤ for Gudder [26], we shall refer to ≤ as the numerical order on B sa (H) and on subalgebras thereof. The "unit interval"and it is the prototype for the more general notion of an effect algebra [8].Let R be a von Neumann algebra and let R sa be the self-adjoint part of R with the numerical order. The system E(is an effect algebra that generalizes the standard effect algebra.By a well-known theorem of S. Sherman [43], R sa is a lattice iff R is commutative. Recall that R is a factor iff its center consists only of scalar multiples of the identity, and R sa is an antilattice iff, for all a, b ∈ R sa , the infimum a ∧ b exists in R sa iff a ≤ b or b ≤ a. By Kadison's antilattice theorem [31], R is a factor iff R sa is an antilattice. Thus, since B(H) is a factor, it follows that B sa (H) is an antilattice under the numerical order.Kadison's antilattice theorem has engendered considerable research on possible alternative orders for B sa (H) and related operator algebras that, optimally, would result in a lattice, or at least in a poset with some latticelike properties. Three (actually two) of these alternative orders are as follows:(1) Drazin's star order ≤ * [6], which was originally defined for so-called proper involution rings, makes sense for a von Neumann algebra R, and so does its restriction to the self-adjoint part R sa of R. The star order on R sa has various equivalent characterizations, one of which is as follows. For A, B ∈ R sa , A ≤ * B iff there exists C ∈ R sa such that AC = 0 and A + C = B. A number of authors, e.g., [3,5,41,45] have studied the star order for various operator algebras and related structures.(2) S.P. Gudder [26] has formulated a so-called logical order for B sa (H) as follows: If A ∈ B sa (H), F (R) is the σ-field of real Borel sets, and P(H) is the lattice under the numerical order of all projections P = P 2 ∈ B sa (H), denote the real observable corresponding to A by P A : F (R) → P(H). Then for A, B ∈ B sa (H), A B iff P A (∆) ≤ P B (∆) for all ∆ ∈ F (R) with 0 / ∈ ∆.

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Section: Introductionmentioning

confidence: 99%

“…A synaptic algebra is lattice ordered if and only if it is commutative [19,Theorem 5.6]. An analogue of Kadison's antilattice theorem [31] is proved for SAs in [20].…”

Section: Introductionmentioning

confidence: 99%