Super quantum measures on effect algebras with the Riesz decomposition properties

2015 ◽  
Vol 56 (10) ◽  
pp. 103509
Author(s):  
Yongjian Xie ◽  
Aili Yang ◽  
Fang Ren
Keyword(s):  
Effect Algebras ◽  
Riesz Decomposition ◽  
Riesz Decomposition Properties ◽  
Decomposition Properties

Intervals of effect algebras and pseudo-effect algebras

2010 ◽  
Vol 60 (5) ◽  
Author(s):  
Ivan Chajda ◽  
Jan Kühr
Keyword(s):  
Effect Algebra ◽  
Effect Algebras ◽  
Pseudo Effect Algebra ◽  
Arbitrary Interval ◽  
Riesz Decomposition ◽  
Riesz Decomposition Properties ◽  
Decomposition Properties

AbstractIt is shown that an arbitrary interval of a pseudo-effect algebra is a pseudo-effect algebra and some results concerning Riesz decomposition properties, compatibilities and states are proved.

scholarly journals Perfect effect algebras are categorically equivalent with Abelian interpolation po-groups

2007 ◽  
Vol 82 (2) ◽  
pp. 183-207 ◽  
Author(s):  
Anatolij Dvurečenskij
Keyword(s):  
Effect Algebra ◽  
Subdirect Product ◽  
Prime Ideals ◽  
Effect Algebras ◽  
Riesz Decomposition Property ◽  
Decomposition Property ◽  
Riesz Decomposition ◽  
The Riesz Decomposition Property

AbstractWe introduce perfect effect algebras and we show that every perfect algebra is an interval in the lexicographical product of the group of all integers with an Abelian directed interpolation po-group. To show this we introduce prime ideals of effect algebras with the Riesz decomposition property (RDP). We show that the category of perfect effect algebras is categorically equivalent to the category of Abelian directed interpolation po-groups. Moreover, we prove that any perfect effect algebra is a subdirect product of antilattice effect algebras with the RDP.

Measures on effect algebras

2019 ◽  
Vol 69 (1) ◽  
pp. 159-170
Author(s):  
Giuseppina Barbieri ◽  
Francisco J. García-Pacheco ◽  
Soledad Moreno-Pulido
Keyword(s):  
Effect Algebras ◽  
Vector Measures ◽  
Finite Dimensional ◽  
Riesz Decomposition ◽  
Nikodym Property ◽  
Additive Measures ◽  
Finite Dimensional Banach Space ◽  
The Riesz Decomposition Property ◽  
Additive Vector ◽  
Additive Vector Measure

Abstract We study measures defined on effect algebras. We characterize real-valued measures on effect algebras and find a class of effect algebras, that include the natural effect algebras of sets, on which σ-additive measures with values in a finite dimensional Banach space are always bounded. We also prove that in effect algebras the Nikodym and the Grothendieck properties together imply the Vitali-Hahn-Saks property, and find an example of an effect algebra verifying the Vitali-Hahn-Saks property but failing to have the Nikodym property. Finally, we define the concept of variation for vector measures on effect algebras proving that in effect algebras verifying the Riesz Decomposition Property, the variation of a finitely additive vector measure is a finitely additive positive measure.

scholarly journals Central elements and Cantor-Bernstein's theorem for pseudo-effect algebras

2003 ◽  
Vol 74 (1) ◽  
pp. 121-144 ◽  
Author(s):  
Anatolij Dvurečenskij
Keyword(s):  
Effect Algebras ◽  
Riesz Decomposition Property ◽  
Bernstein Theorem ◽  
Riesz Decomposition ◽  
Mv Algebra ◽  
Central Elements ◽  
Partial Algebras ◽  
Left And Right ◽  
The Riesz Decomposition Property ◽  
Mv Algebras

AbstractPseudo-effect algebras are partial algebras (E; +, 0, 1) with a partially defined addition + which is not necessary commutative and with two complements, left and right ones. We define central elements of a pseudo-effect algebra and the centre, which in the case of MV-algebras coincides with the set of Boolean elements and in the case of effect algebras with the Riesz decomposition property central elements are only characteristic elements. If E satisfies general comparability, then E is a pseudo MV-algebra. Finally, we apply central elements to obtain a variation of the Cantor-Bernstein theorem for pseudo-effect algebras.

scholarly journals Blocks of homogeneous effect algebras

2001 ◽  
Vol 64 (1) ◽  
pp. 81-98 ◽  
Author(s):  
Gejza Jenča
Keyword(s):  
Effect Algebra ◽  
Effect Algebras ◽  
Orthomodular Lattices ◽  
Riesz Decomposition Property ◽  
Decomposition Property ◽  
New Class ◽  
Riesz Decomposition ◽  
Partial Algebras ◽  
Homogeneous Effect Algebra ◽  
The Riesz Decomposition Property

Effect algebras, introduced by Foulis and Bennett in 1994, are partial algebras which generalise some well known classes of algebraic structures (for example orthomodular lattices, MV algebras, orthoalgebras et cetera). In the present paper, we introduce a new class of effect algebras, calledhomogeneous effect algebras. This class includes orthoalgebras, lattice ordered effect algebras and effect algebras satisfying the Riesz decomposition property. We prove that every homogeneous effect algebra is a union of its blocks, which we define as maximal sub-effect algebras satisfying the Riesz decomposition property. This generalizes a recent result by Riec˘anová, in which lattice ordered effect algebras were considered. Moreover, the notion of a block of a homogeneous effect algebra is a generalisation of the notion of a block of an orthoalgebra. We prove that the set of all sharp elements in a homogeneous effect algebraEforms an orthoalgebraEs. Every block ofEsis the centre of a block ofE. The set of all sharp elements in the compatibility centre ofEcoincides with the centre ofE. Finally, we present some examples of homogeneous effect algebras and we prove that for a Hilbert space ℍ with dim (ℍ) > 1, the standard effect algebra ℰ(ℍ) of all effects in ℰ is not homogeneous.

The Structure of Positive Elements for C*-Algebras with Real Rank Zero

1997 ◽  
Vol 08 (03) ◽  
pp. 383-405 ◽  
Author(s):  
Francesc Perera
Keyword(s):  
Grothendieck Group ◽  
Real Rank ◽  
Real Rank Zero ◽  
Rank Zero ◽  
C Algebra ◽  
C Algebras ◽  
Positive Elements ◽  
Riesz Decomposition ◽  
Riesz Decomposition Properties ◽  
Rank One

In this paper we give a representation theorem for the Cuntz monoid S(A) of a σ-unital C*-algebra A with real rank zero and stable rank one, which allows to prove several Riesz decomposition properties on the monoid. As a consequence, it is proved that the comparability conditions (FCQ), stable (FCQ) and (FCQ+) are equivalent for simple C*-algebras with real rank zero. It is also shown that the Grothendieck group [Formula: see text] of S(A) is a Riesz group, and lattice-ordered under some additional assumptions on A.

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