2,030 research outputs found
States on pseudo effect algebras and integrals
We show that every state on an interval pseudo effect algebra satisfying
some kind of the Riesz Decomposition Properties (RDP) is an integral through a
regular Borel probability measure defined on the Borel -algebra of a
Choquet simplex . In particular, if satisfies the strongest type of
(RDP), the representing Borel probability measure can be uniquely chosen to
have its support in the set of the extreme points of $K.
Recasting the Elliott conjecture
Let A be a simple, unital, exact, and finite C*-algebra which absorbs the
Jiang-Su algebra Z tensorially. We prove that the Cuntz semigroup of A admits a
complete order embedding into an ordered semigroup obtained from the Elliott
invariant in a functorial manner. We conjecture that this embedding is an
isomorphism, and prove the conjecture in several cases. In these same cases --
Z-stable algebras all -- we prove that the Elliott conjecture in its strongest
form is equivalent to a conjecture which appears much weaker. Outside the class
of Z-stable algebras, this weaker conjecture has no known counterexamples, and
it is plausible that none exist. Thus, we reconcile the still intact principle
of Elliott's classification conjecture -- that K-theoretic invariants will
classify separable and nuclear C*-algebras -- with the recent appearance of
counterexamples to its strongest concrete form.Comment: 28 pages; several typos corrected, Lemma 3.4 added; to appear in
Math. An
Finitely presented lattice-ordered abelian groups with order-unit
Let be an -group (which is short for ``lattice-ordered abelian
group''). Baker and Beynon proved that is finitely presented iff it is
finitely generated and projective. In the category of {\it unital}
-groups---those -groups having a distinguished order-unit
---only the -direction holds in general. Morphisms in
are {\it unital -homomorphisms,} i.e., hom\-o\-mor\-phisms
that preserve the order-unit and the lattice structure. We show that a unital
-group is finitely presented iff it has a basis, i.e., is
generated by an abstract Schauder basis over its maximal spectral space. Thus
every finitely generated projective unital -group has a basis . As a partial converse, a large class of projectives is constructed from
bases satisfying . Without using the
Effros-Handelman-Shen theorem, we finally show that the bases of any finitely
presented unital -group provide a direct system of simplicial
groups with 1-1 positive unital homomorphisms, whose limit is
Lexicographic Effect Algebras
In the paper we investigate a class of effect algebras which can be
represented in the form of the lexicographic product \Gamma(H\lex G,(u,0)),
where is an Abelian unital po-group and is an Abelian directed
po-group. We study algebraic conditions when an effect algebra is of this form.
Fixing a unital po-group , the category of strong -perfect effect
algebra is introduced and it is shown that it is categorically equivalent to
the category of directed po-group with interpolation. We show some
representation theorems including a subdirect product representation by
antilattice lexicographic effect algebras
The Lattice and Simplex Structure of States on Pseudo Effect Algebras
We study states, measures, and signed measures on pseudo effect algebras with
some kind of the Riesz Decomposition Property, (RDP). We show that the set of
all Jordan signed measures is always an Abelian Dedekind complete -group.
Therefore, the state space of the pseudo effect algebra with (RDP) is either
empty or a nonempty Choquet simplex or even a Bauer simplex. This will allow
represent states on pseudo effect algebras by standard integrals
Pseudo MV-algebras and Lexicographic Product
We study algebraic conditions when a pseudo MV-algebra is an interval in the
lexicographic product of an Abelian unital -group and an -group
that is not necessary Abelian. We introduce -perfect pseudo MV-algebras
and strong -perfect pseudo MV-algebras, the latter ones will have a
representation by a lexicographic product. Fixing a unital -group
, the category of strong -perfect pseudo MV-algebras is
categorically equivalent to the category of -groups.Comment: arXiv admin note: text overlap with arXiv:1304.074
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