scholarly journals The three types of normal sequential effect algebras

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 378
Author(s):  
Abraham Westerbaan ◽  
Bas Westerbaan ◽  
John van de Wetering
Keyword(s):  
Boolean Algebra ◽  
Effect Algebra ◽  
Unit Interval ◽  
Sequential Effect ◽  
Spectral Theorem ◽  
Von Neumann ◽  
Square Roots ◽  
Almost Convex ◽  
Sequential Product ◽  
Directed Sets

A sequential effect algebra (SEA) is an effect algebra equipped with a sequential product operation modeled after the Lüders product (a,b)↦aba on C∗-algebras. A SEA is called normal when it has all suprema of directed sets, and the sequential product interacts suitably with these suprema. The effects on a Hilbert space and the unit interval of a von Neumann or JBW algebra are examples of normal SEAs that are in addition convex, i.e. possess a suitable action of the real unit interval on the algebra. Complete Boolean algebras form normal SEAs too, which are convex only when 0=1.We show that any normal SEA E splits as a direct sum E=Eb⊕Ec⊕Eac of a complete Boolean algebra Eb, a convex normal SEA Ec, and a newly identified type of normal SEA Eac we dub purely almost-convex.Along the way we show, among other things, that a SEA which contains only idempotents must be a Boolean algebra; and we establish a spectral theorem using which we settle for the class of normal SEAs a problem of Gudder regarding the uniqueness of square roots. After establishing our main result, we propose a simple extra axiom for normal SEAs that excludes the seemingly pathological a-convex SEAs. We conclude the paper by a study of SEAs with an associative sequential product. We find that associativity forces normal SEAs satisfying our new axiom to be commutative, shedding light on the question of why the sequential product in quantum theory should be non-associative.

scholarly journals Orthomodular generalizations of homogeneous Boolean algebras

1973 ◽  
Vol 15 (1) ◽  
pp. 94-104 ◽  
Author(s):  
C. H. Randall ◽  
M. F. Janowitz ◽  
D. J. Foulis
Keyword(s):  
Boolean Algebra ◽  
Dense Subset ◽  
Unit Interval ◽  
Boolean Algebras ◽  
Borel Sets ◽  
History Of ◽  
Atomic Boolean Algebra

It is well known that the so-called reduced Borel algebra, that is, the Boolean algebra of all Borel subsets of the unit interval modulo the meager Borel sets of this interval, can be abstractly characterized as a complete, totally non-atomic Boolean algebra containing a countable join dense subset. (For an indication of the history of this result, see, for example, ([2], p. 483, footnote 12.) From this characterization, it easily follows that the reduced Borel algebra B is “homogeneous” in the sense that every non-trivial in B is isomorphic to B.

Rényi Entropy and Rényi Divergence in Sequential Effect Algebra

2020 ◽  
Vol 27 (02) ◽  
pp. 2050008
Author(s):  
Zahra Eslami Giski
Keyword(s):  
Shannon Entropy ◽  
Effect Algebra ◽  
Sequential Effect ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Numerical Examples ◽  
Basic Properties ◽  
Rényi Divergence ◽  
Leibler Divergence ◽  
Entropy Measures

The aim of this study is to extend the results concerning the Shannon entropy and Kullback–Leibler divergence in sequential effect algebra to the case of Rényi entropy and Rényi divergence. For this purpose, the Rényi entropy of finite partitions in sequential effect algebra and its conditional version are proposed and the basic properties of these entropy measures are derived. In addition, the notion of Rényi divergence of a partition in sequential effect algebra is introduced and the basic properties of this quantity are studied. In particular, it is proved that the Kullback–Leibler divergence and Shannon’s entropy of partitions in a given sequential effect algebra can be obtained as limits of their Rényi divergence and Rényi entropy respectively. Finally, to illustrate the results, some numerical examples are presented.

scholarly journals Not each sequential effect algebra is sharply dominating

2009 ◽  
Vol 373 (20) ◽  
pp. 1708-1712 ◽  
Author(s):  
Jun Shen ◽  
Junde Wu
Keyword(s):  
Effect Algebra ◽  
Sequential Effect

scholarly journals An introduction of logical entropy on sequential effect algebra

2017 ◽  
Vol 28 (5) ◽  
pp. 928-937 ◽  
Author(s):  
Zahra Eslami Giski ◽  
Abolfazl Ebrahimzadeh
Keyword(s):  
Effect Algebra ◽  
Sequential Effect ◽  
Logical Entropy

scholarly journals TYPE DECOMPOSITION OF A PSEUDOEFFECT ALGEBRA

2010 ◽  
Vol 89 (3) ◽  
pp. 335-358 ◽  
Author(s):  
DAVID J. FOULIS ◽  
SYLVIA PULMANNOVÁ ◽  
ELENA VINCEKOVÁ
Keyword(s):  
Effect Algebra ◽  
Von Neumann Algebra ◽  
Von Neumann ◽  
Pseudoeffect Algebras ◽  
Noncommutative Version ◽  
Pseudoeffect Algebra ◽  
Determining Set ◽  
Direct Decompositions ◽  
Type Decomposition ◽  
Unsharp Observables

AbstractEffect algebras, which generalize the lattice of projections in a von Neumann algebra, serve as a basis for the study of unsharp observables in quantum mechanics. The direct decomposition of a von Neumann algebra into types I, II, and III is reflected by a corresponding decomposition of its lattice of projections, and vice versa. More generally, in a centrally orthocomplete effect algebra, the so-called type-determining sets induce direct decompositions into various types. In this paper, we extend the theory of type decomposition to a (possibly) noncommutative version of an effect algebra called a pseudoeffect algebra. It has been argued that pseudoeffect algebras constitute a natural structure for the study of noncommuting unsharp or fuzzy observables. We develop the basic theory of centrally orthocomplete pseudoeffect algebras, generalize the notion of a type-determining set to pseudoeffect algebras, and show how type-determining sets induce direct decompositions of centrally orthocomplete pseudoeffect algebras.

scholarly journals Cartan Triples

2019 ◽  
Author(s):  
Allan P Donsig ◽  
Adam H Fuller ◽  
David R Pitts
Keyword(s):  
Inverse Semigroup ◽  
Von Neumann Algebra ◽  
Inverse Semigroups ◽  
Spectral Theorem ◽  
Von Neumann ◽  
Support Sets ◽  
One To One

Abstract We introduce the class of Cartan triples as a generalization of the notion of a Cartan MASA in a von Neumann algebra. We obtain a one-to-one correspondence between Cartan triples and certain Clifford extensions of inverse semigroups. Moreover, there is a spectral theorem describing bimodules in terms of their support sets in the fundamental inverse semigroup and, as a corollary, an extension of Aoi’s theorem to this setting. This context contains that of Fulman’s generalization of Cartan MASAs and we discuss his generalization in an appendix.

scholarly journals A Note of Filters in Effect Algebras

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Biao Long Meng ◽  
Xiao Long Xin
Keyword(s):  
Boolean Algebra ◽  
Orthomodular Lattice ◽  
Effect Algebra ◽  
Effect Algebras ◽  
Lattice Filter ◽  
Mv Algebra ◽  
Mv Algebras

We investigate relations of the two classes of filters in effect algebras (resp., MV-algebras). We prove that a lattice filter in a lattice ordered effect algebra (resp., MV-algebra) does not need to be an effect algebra filter (resp., MV-filter). In general, in MV-algebras, every MV-filter is also a lattice filter. Every lattice filter in a lattice ordered effect algebra is an effect algebra filter if and only if is an orthomodular lattice. Every lattice filter in an MV-algebra is an MV-filter if and only if is a Boolean algebra.

scholarly journals The Center Of A Generalized Effect Algebra

2014 ◽  
Vol 47 (1) ◽  
pp. 1-21 ◽  
Author(s):  
D. J. Foulis ◽  
S. Pulmannová
Keyword(s):  
Boolean Algebra ◽  
Direct Summand ◽  
Effect Algebra ◽  
Complete Boolean Algebra ◽  
Cover System ◽  
Direct Sum ◽  
Generalized Effect Algebra

AbstractIn this article, we study the center of a generalized effect algebra (GEA), relate it to the exocenter, and in case the GEA is centrally orthocomplete (a COGEA), relate it to the exocentral cover system. Our main results are that the center of a COGEA is a complete boolean algebra and that a COGEA decomposes uniquely as the direct sum of an effect algebra (EA) that contains the center of the COGEA and a complementary direct summand in which no nonzero direct summand is an EA.

The center of an orthologic

1972 ◽  
Vol 37 (4) ◽  
pp. 641-645 ◽  
Author(s):  
Barbara Jeffcott
Keyword(s):  
Quantum Mechanics ◽  
Boolean Algebra ◽  
Quantum Logic ◽  
Decisive Role ◽  
Orthomodular Poset ◽  
Fundamental Relation ◽  
Von Neumann ◽  
Modular Ortholattice ◽  
Probability And Statistics ◽  
Kolmogorov Theory

Since 1933, when Kolmogorov laid the foundations for probability and statistics as we know them today [1], it has been recognized that propositions asserting that such and such an event occurred as a consequence of the execution of a particular random experiment tend to band together and form a Boolean algebra. In 1936, Birkhoff and von Neumann [2] suggested that the so-called logic of quantum mechanics should not be a Boolean algebra, but rather should form what is now called a modular ortholattice [3]. Presumably, the departure from Boolean algebras encountered in quantum mechanics can be attributed to the fact that in quantum mechanics, one must consider more than one physical experiment, e.g., an experiment measuring position, an experiment measuring charge, an experiment measuring momentum, etc., and, because of the uncertainty principle, these experiments need not admit a common refinement in terms of which the Kolmogorov theory is directly applicable.Mackey's Axioms I–VI for quantum mechanics [4] imply that the logic of quantum mechanics should be a σ-orthocomplete orthomodular poset [5]. Most contemporary practitioners of quantum logic seem to agree that a quantum logic is (at least) an orthomodular poset [6], [7], [8], [9], [10] or some variation thereof [11]. P. D. Finch [12] has shown that every completely orthomodular poset is the logic arising from sets of Boolean logics, where these sets have a structure similar to the structures generally given to quantum logic. In all of these versions of quantum logic, a fundamental relation, the relation of compatibility or commutativity, plays a decisive role.

Some Examples of Complemented Modular Lattices

1962 ◽  
Vol 5 (2) ◽  
pp. 111-121 ◽  
Author(s):  
G. Grätzer ◽  
Maria J. Wonenburger
Keyword(s):  
Boolean Algebra ◽  
Modular Lattice ◽  
Complete Boolean Algebra ◽  
Modular Lattices ◽  
Additive Property ◽  
Von Neumann ◽  
Continuous Geometry ◽  
Definition Of ◽  
Homogeneous Basis

Let L be a complemented, χ-complete modular lattice. A theorem of Amemiya and Halperin (see [l], Theorem 4.3) asserts that if the intervals [O, a] and [O, b], a, bεL, are upper χ-continuous then [O, a∪b] is also upper χ-continuous. Roughly speaking, in L upper χ-continuity is additive. The following question arises naturally: is χ-completeness an additive property of complemented modular lattices? It follows from Corollary 1 to Theorem 1 below that the answer to this question is in the negative.A complemented modular lattice is called a Von Neumann geometry or continuous geometry if it is complete and continuous. In particular a complete Boolean algebra is a Von Neumann geometry. In any case in a Von Neumann geometry the set of elements which possess a unique complement form a complete Boolean algebra. This Boolean algebra is called the centre of the Von Neumann geometry. Theorem 2 shows that any complete Boolean algebra can be the centre of a Von Neumann geometry with a homogeneous basis of order n (see [3] Part II, definition 3.2 for the definition of a homogeneous basis), n being any fixed natural integer.

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