scholarly journals Positive bases in ordered subspaces with the Riesz decomposition property

2006 ◽  
Vol 174 (3) ◽  
pp. 233-253 ◽  
Author(s):  
Vasilios Katsikis ◽  
Ioannis A. Polyrakis
Keyword(s):  
Riesz Decomposition Property ◽  
Decomposition Property ◽  
Riesz Decomposition ◽  
Positive Bases ◽  
The Riesz Decomposition Property

C* -Algebras of Real Rank Zero Whose K0's are not Riesz Groups

1996 ◽  
Vol 39 (4) ◽  
pp. 429-437 ◽  
Author(s):  
K. R. Goodearl
Keyword(s):  
Real Rank ◽  
Riesz Decomposition Property ◽  
Real Rank Zero ◽  
Decomposition Property ◽  
Rank Zero ◽  
C Algebras ◽  
Finite Dimensional ◽  
Riesz Decomposition ◽  
Stably Finite ◽  
The Riesz Decomposition Property

AbstractExamples are constructed of stably finite, imitai, separable C* -algebras A of real rank zero such that the partially ordered abelian groups K0(A) do not satisfy the Riesz decomposition property. This contrasts with the result of Zhang that projections in C* -algebras of real rank zero satisfy Riesz decomposition. The construction method also produces a stably finite, unital, separable C* -algebra of real rank zero which has the same K-theory as an approximately finite dimensional C*-algebra, but is not itself approximately finite dimensional.

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.

scholarly journals The spectrum of an R-homomorphism

1977 ◽  
Vol 23 (1) ◽  
pp. 42-45 ◽  
Author(s):  
A. W. Wickstead
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Real Banach Space ◽  
Point Spectrum ◽  
Positive Linear Operator ◽  
Riesz Decomposition Property ◽  
Decomposition Property ◽  
Approximate Point Spectrum ◽  
Riesz Decomposition ◽  
The Riesz Decomposition Property

AbstractLet E be a real Banach space ordered by a closed, normal and generating cone. Suppose also that the order induced on E has the Riesz decomposition property. It is shown that if T:E → E is a positive linear operator with the property that y, z, a ∈ E with a ≧ Ty, Tz implies there is x ∈ E with x ≧ y, z and a ≧ Tx then the approximate point spectrum and spectrum of T are cyclic subsets of the complex plane. That is, if α = |α|γ lies in one of these sets then so does |α|γk for all integers k.

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.

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