Hewitt and Lebesgue Decompositions of States on Orthomodular Posets1

Journal of Mathematical Analysis and Applications 255, 74–104 (2001) doi:10.1006/jmaa.2000.7160, available online at http://www.idealibrary.com on Yosida–Hewitt and Lebesgue Decompositions of States on Orthomodular Posets1 Anna De Simone Dip. di Matematica e Applicazioni “R. Caccioppoli,” Universita` degli Studi di Napoli “Federico II,” Complesso Monte S. Angelo, Via Cintia, 80126 Napoli, Italy E-mail: [email protected] and Mirko Navara Center for Machine Perception, Faculty of Electrical Engineering, Czech Technical University, Technicka´ 2, 166 27 Praha, Czech Republic E-mail: [email protected] Submitted by A. Di Nola Received December 13, 1999 Orthomodular posets are usually used as event structures of quantum mechanical systems. The states of the systems are described by probability measures (also called states) on it. It is well known that the family of all states on an orthomodular poset is a convex set, compact with respect to the product topology. This suggests using geometrical results to study its structure. In this line, we deal with the problem of the decomposition of states on orthomodular posets with respect to a given face of the state space. For particular choices of this face, we obtain, e.g., Lebesgue- type and Yosida–Hewitt decompositions as special cases. Considering, in particular, the problem of existence and uniqueness of such decompositions, we generalize to this setting numerous results obtained earlier only for orthomodular lattices and orthocomplete orthomodular posets. © 2001 Academic Press Key Words: face of a convex set; state; probability measure; orthomodular poset; Yosida–Hewitt decomposition; Lebesgue decomposition; filtering set; filtering function; heredity. 1 The authors gratefully acknowledge the support of Project VS96049 of the Czech Ministry of Education and Grant 201/00/0331 of the Grant Agency of the Czech Republic. The research of the first author was partially supported by Ministero dell’Università e della Ricerca Scien- tifica e Tecnologica (Italy). 74 0022-247X/01 $35.00 Copyright © 2001 by Academic Press All rights of reproduction in any form reserved. orthomodular posets 75 1. INTRODUCTION Some of the classical decomposition theorems, originally stated for measures on Boolean algebras, have been extended to the case of nondistributive structures—orthomodular posets (OMPs)—giving also interesting applications to functional analysis (think, for example, of the Yosida–Hewitt decomposition theorem [28] stated by Aarnes in [1] for the case of the orthomodular lattice of closed subspaces of a Hilbert space). We investi- gate this topic from a geometrical point of view: we decompose finitely additive probability measures (states) defined on an OMP with respect to a face of its state space (which is a compact convex subset of a locally convex Hausdorff topological linear space). This approach is sufficiently general to give Lebesgue-type and Yosida–Hewitt decompositions as particular cases. We also deal with some related questions and prove the validity in general OMPs of many results which are known to hold in less general structures. In extending decomposition results from the case of Boolean algebras to the more general case of OMPs, it is possible to obtain theorems which state the existence of decomposition, but not its uniqueness [7, 8, 19], and in fact in these structures uniqueness fails to hold in general (see Examples 3.13 and 4.3). Only in some particular cases the uniqueness is proved; see, e.g., [27] for the case of valuations, and [2, 3], where it is treated for the case of functions whose kernel is a p-ideal. Hence it is desirable to have some conditions which are sufficient for uniqueness. The paper [16] contributed to the study of the Yosida–Hewitt decomposition by showing that in some sense the lack of uniqueness is “natural” for measures defined on nondistributive structures. The research of the uniqueness of the Yosida–Hewitt decomposition was initiated in [21] and generalized in [9]. In this paper we first study a general type of decomposition, showing that it covers the Lebesgue and Yosida–Hewitt decompositions as particular cases. We then use the concept of filtering states (Definition 7.1) for comparing Yosida–Hewitt decomposition with a type of decomposition introduced by Ruttimann¨ in [21]. The first is known to exist for any OMP; the second does not exist in general, but, if this is the case, then it is unique. When they coincide (this happens exactly when each state admits aRuttimann¨ decomposition), we obtain the existence and uniqueness of such a decomposition (Theorem 8.1). We then introduce two concepts of “heredity” in OMPs (Definitions 8.6 and 9.5), less general than the previ- ous coincidence, but still strong enough to ensure the uniqueness of Yosida– Hewitt decomposition (Corollary 9.12). We observe that this can be applied to the particular case of Boolean algebras. The result is a modified version of [9, Theorem 2]—it is stated for OMPs (in this sense it is more general) and by using faces (in this sense it is less general). 76 de simone and navara Examples are important not only to support the reader’s intuition, but also to show when the uniqueness of decomposition follows from simple geometric properties of the state space and when it is more involved. We therefore include several of them, providing answers to natural questions arising in this context. The paper is organized as follows: Section 2 contains all the notions used in the sequel. Section 3 is devoted to the description of D-decomposition (using faces); the results are applied in Section 4 to the D-decomposition of states on OMPs. The general case is followed by several particular cases—Lebesgue decomposition (Section 5) and Yosida–Hewitt decomposition (Section 6). Another decomposition to filtering states and completely additive states (Ruttimann¨ decomposition) is studied in Section 7; its relation to the Yosida–Hewitt decomposition is described in Section 8. Section 9 presents an alternative approach—decompositions using weakly filtering states—and a “graphic” summary of the results (Fig. 4). 2. BASIC NOTIONS In this section, we present basic notions and results concerning orthomodular posets. These are generalizations of Boolean algebras which admit the phenomenon of noncompatibility: the existence of pairs of elements which can be observed separately, but non-simultaneously. This makes orthomodular posets a well-motivated structure for the description of events of quantum mechanical systems where noncompatibility is a characteristic feature. Consider L ≤ 0 1 , where L is a set, ≤ is a binary relation on L, 0 and 1 are two distinct elements of L and is a function from L to L. We say that L ≤ 0 1 is an orthomodular poset (OMP) if the following conditions are satisfied: (i) L ≤ is a partially ordered set with a least element 0 and a greatest element 1, (ii) L → L is a decreasing function such that p = p and p ∧ p = 0 for all p ∈ L, (iii) if p q ∈ L with p q orthogonal (i.e., p ≤ q, in symbols p ⊥ q), then p ∨ q exists in L, (iv) if p q ∈ L with p ≤ q, then q = p ∨p ∧ q (orthomodular law). For brevity we will only use L to denote the OMP L ≤ 0 1 . If, in addition, L ≤ is a lattice then L is called an orthomodular lattice (OML). A distributive orthomodular lattice is a Boolean algebra. A very important example of a nondistributive orthomodular lattice is the lattice of orthomodular posets 77 FIG. 1. Hasse diagram of the OML MO2 (Example 2.1). projections in a Hilbert space or, more generally, in a von Neumann algebra. Numerous examples of (finite or infinite) OMPs can be found, e.g., in [11, 15]. A subset M of L is called orthogonal if its elements are pairwise orthogonal. If all orthogonal subsets of L have supremum in L, then L is called orthocomplete.ForM ⊆ L and p ∈ L, we use the notation M ≤ p (resp. M ⊥ p) to say that each element in M is less than or equal to p (resp. orthogonal to p). Example 2.1. The OML MO2 is described by its Hasse diagram in Fig. 1. Its elements are 0 1ppqq. For two OMPs K L, a mapping f : K → L is called a homomorphism if (1) f 0=0, (2) if p ∈ L, then f p=f p, (3) if p q ∈ L, p ⊥ q, then f p ∨ q=f p∨f q. If, moreover, f is injective, surjective, and f −1 is also a homomorphism, then f is called an isomorphism. The following notion will play an important role in the sequel. Definition 2.2. A subset I of an OMP L is called filtering (join dense) if ∀p ∈ L\0∃q ∈ I\0q ≤ p. Example 2.3. Let λ be the Lebesgue measure on the algebra L of Lebesgue measurable subsets of 0 1 and let L be its quotient with respect to the set =p ∈ L λp=0. For any subset F of L, denote by F the corresponding set in L. Note that the set F1 =xx ∈0 1 is filtering in L, while F1 is not filtering in L (in fact F1 only contains the zero −n element of L). On the other hand, F2 =q ∈ L λq=2 for some n ∈ is not filtering in L, while F2 is filtering in L. Lemma 2.4. If M is a maximal orthogonal subset of a filtering set F, then M = 1. 78 de simone and navara Proof. If p ∈ L is an upper bound of M different from 1, then from the relation p = 0 it follows that F contains a nonzero element q ≤ p.

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