Archimedeanness and the MacNeille completion of pseudoeffect algebras and po-groups

Pseudoeffect (PE-) algebras are partial algebras differing from effect algebras in that they need not satisfy the commutativity assumption. PE-algebras typically arise from intervals of po-groups; this applies in particular to all those which satisfy a certain Riesz property.In this paper, we discuss the property of archimedeanness for PE-algebras on the one hand and for po-groups on the other hand. We prove that under the assumption of suphomogeneity, archimedeanness holds for a PE-algebra with the Riesz property if and only if it holds for its representing group. The algebra is in that case commutative. This result is established by using the technique of MacNeille completion. We give the exact condition for this completion to exist, and we clearly exhibit the role played by archimedeanness and by sup-homogeneity.     

Matrix algebras with Multiplicative Decomposition Property

We consider the Multiplicative Decomposition Property (the multiplicative analogue of the Riesz Decomposition Property) for entry-wise ordered matrix algebras over the reals. We characterize for which matrix pairs such a decomposition exists and use this result to present necessary and sufficient conditions for a matrix algebra to enjoy the decomposition property by all its positive matrix pairs. Thus we describe all matrix algebras with the decomposition property.     

Effect Algebras as Presheaves on Finite Boolean Algebras

For an effect algebra A, we examine the category of all morphisms from finite Boolean algebras into A. This category can be described as a category of elements of a presheaf R(A) on the category of finite Boolean algebras. We prove that some properties (being an orthoalgebra, the Riesz decomposition property, being a Boolean algebra) of an effect algebra A can be characterized in terms of some properties of the category of elements of the presheaf R(A). We prove that the tensor product of effect algebras arises as a left Kan extension of the free product of finite Boolean algebras along the inclusion functor. The tensor product of effect algebras can be expressed by means of the Day convolution of presheaves on finite Boolean algebras.     

On Cantor-Bernstein type theorems in Riesz spaces

We generalize the main result of [21] to Riesz spaces. Let X and Y be Riesz spaces with σ-complete Boolean algebras of projection bands. If X and Y are each Riesz isomorphic to a projection band of the other space then the spaces are Riesz isomorphic. As an application of the above theorem we give an example of non-Riesz isomorphic Banach lattices such that: (1) their order (= topological) duals are Riesz isomorphic and (2) each of them is Riesz isomorphic to a projection band of the other one.     

Quasicompact and Riesz endomorphisms of Banach algebras

Let B be a unital commutative semi-simple Banach algebra. We study endomorphisms of B which are also quasicompact operators or Riesz operators. Clearly compact and power compact endomorphisms are Riesz and hence quasicompact. Several general theorems about quasicompact endomorphisms are proved, and these results are then applied to the question of when quasicompact or Riesz endomorphisms of certain algebras are necessarily power compact.     

On the regularity of homomorphisms between Riesz subalgebras of Lr(X), II

We extend and improve our earlier results on automatic regularity of continuous algebra homomorphisms between Riesz algebras of regular operators.     

Riesz ideals in generalized effect algebras and in their unitizations

We show that quotients of generalized effect algebras by Riesz ideals preserve some important special properties, e.g., homogeneity and hereditary Riesz decomposition properties; moreover, quotients of generalized orthoalgebras, weak generalized orthomodular posets, generalized orthomodular lattices and generalized MV-algebras with respect to Riesz ideals belong to the same class. We give a necessary and sufficient condition under which a Riesz ideal I of a generalized effect algebra P is a Riesz ideal also in the unitization E of P. We also study relations between Riesz ideals and central elements in GEAs and in their unitizations. In the last section, we demonstrate the notion of Riesz ideals by some illustrative examples. Received June 28, 2005; accepted in final form January 23, 2007.     

Congruences and Ideals of Effect Algebras

This paper is devoted to a general investigation of congruences and ideals in effect algebras. One of our main results is the existence of an order isomorphism between Riesz congruences and Riesz ideals. We also answer an open question of Dvureenskij and Pulmannová by showing that an ideal is a Riesz ideal if and only if it is closed under generalized Sasaki projections.     

Partial algebras for Łukasiewicz logics and its extensions

It is a well-known fact that MV-algebras, the algebraic counterpart of Łukasiewicz logic, correspond to a certain type of partial algebras: lattice-ordered effect algebras fulfilling the Riesz decomposition property. The latter are based on a partial, but cancellative addition, and we may construct from them the representing ℓ-groups in a straightforward manner. In this paper, we consider several logics differing from Łukasiewicz logics in that they contain further connectives: the PŁ-, PŁ'-, PŁ'_△-, and ŁΠ-logics. For all their algebraic counterparts, we characterise the corresponding type of partial algebras. We moreover consider the representing f-rings. All in all, we get three-fold correspondences: the total algebras - the partial algebras - the representing rings.     

COMPLEX INTERPOLATION OF NONCOMMUTATIVE HARDY SPACES ASSOCIATED WITH SEMIFINITE VON NEUMANN ALGEBRAS

We proved a complex interpolation theorem of noncommutative Hardy spaces associated with semi-finite von Neumann algebras and extend the Riesz type factorization to the semi-finite case.     

Decomposition of ℓ-group-valued measures

We deal with decomposition theorems for modular measures µ: L → G defined on a D-lattice with values in a Dedekind complete ?-group. Using the celebrated band decomposition theorem of Riesz in Dedekind complete ?-groups, several decomposition theorems including the Lebesgue decomposition theorem, the Hewitt-Yosida decomposition theorem and the Alexandroff decomposition theorem are derived. Our main result—also based on the band decomposition theorem of Riesz—is the Hammer-Sobczyk decomposition for ?-group-valued modular measures on D-lattices. Recall that D-lattices (or equivalently lattice ordered effect algebras) are a common generalization of orthomodular lattices and of MV-algebras, and therefore of Boolean algebras. If L is an MV-algebra, in particular if L is a Boolean algebra, then the modular measures on L are exactly the finitely additive measures in the usual sense, and thus our results contain results for finitely additive G-valued measures defined on Boolean algebras.     

Quantum logics with the Riesz Interpolation Property

We study the quantum logics which satisfy the Riesz Interpolation Property. We call them the RIP logics. We observe that the class of RIP logics is considerable large—it contains all lattice quantum logics and, also, many (infinite) non‐lattice ones. We then find out that each RIP logic can be enlarged to an RIP logic with a preassigned centre. We continue, showing that the “nearly” Boolean RIP logics must be Boolean algebras. In a somewhat surprising contrast to this, we finally show that the attempt for the σ‐complete formulation of this result fails: We show by constructing an example that there is a non‐Boolean nearly Boolean σ‐RIP logic. As a result, there are interesting σ‐RIP logics which are intrinsically close to Boolean σ‐algebras. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Extensions of partially ordered partial abelian monoids

The notion of a partially ordered partial abelian monoid is introduced and extensions of partially ordered abelian monoids by partially ordered abelian groups are studied. Conditions for the extensions to exist are found. The cases when both the above mentioned structures have the Riesz decomposition property, or are lattice ordered, are treated. Some applications to effect algebras and MV-algebras are shown.     

Quotients of partial abelian monoids

A hierarchy of partial abelian structures is considered. In an order of decreasing generality, these structures include partial abelian monoids (PAM), cancellative PAMs (CPAM), effect algebras (or D-posets), orthoalgebras, orthomodular posets (OMP) and orthomodular lattices (OML). If P is a PAM, the concepts of a congruence on P and a quotient P are defined. Similar definitions are given for quotients of higher level categories in the hierarchy. The notion of a Riesz ideal I on a CPAM P is defined and it is shown that I generates a congruence on P. The corresponding quotients P/I for categories in the hierarchy are studied. It is shown that a subset I of an OML is a Riesz ideal if and only if I is a p-ideal. Moreover, for effect algebras, we show that congruences generated by Riesz ideals are precisely those that are given by a perspectivity. The paper includes a large number of counterexamples and examples that illustrate various concepts. Received April 14, 1997; accepted in final form January 19, 1998.     

When the lexicographic product of two po-groups has the Riesz decomposition property

We study conditions when a certain type of the Riesz Decomposition Property (RDP for short) holds in the lexicographic product of two po-groups. Defining two important properties of po-groups, we extend known situations showing that the lexicographic product satisfies RDP or even \({{\rm RDP}_1}\), a stronger type of RDP. We recall that a very strong type of RDP, \({{\rm RDP}_2}\), entails that the group is lattice ordered. RDP's of the lexicographic products are important for the study of lexicographic pseudo effect algebras, or perfect types of pseudo MV-algebras and pseudo effect algebras, where infinitesimal elements play an important role both for algebras as well as for the first order logic of valid but not provable formulas.     

Banach-Stone theorems and Riesz algebras

Let X, Y be compact Hausdorff spaces and let E, F be both Banach lattices and Riesz algebras. In this paper, the following main result shall be proved: If F has no zero-divisor and there exists a Riesz algebraic isomorphism such that Φ(f) has no zero if f has none, then X is homeomorphic to Y and E is Riesz algebraically isomorphic to F.     

Approximate Radon-Nikodým representations on Riesz algebras

For the finitely additive case approximate Radon-Nikodym representations are obtained in the setting of Riesz algebras, complementing and generalizing results of Bochner [4], Fefferman [10], de Amo, Chitescu and Díaz Carrillo [1]. Furthermore the possible relations between the various spaces which appear here are given, answering a question of [1]. Various examples show that our results are sharp, there also the class of approximately representable functionals is explicitly characterized, a partial answer to another question of [1]. Finally some open questions are listed.     

The Banach-Stone theorem revisited

Let X and Y be compact Hausdorff spaces, and E and F be locally solid Riesz spaces. If π:C(X,E)→C(Y,F) is a 1-biseparating Riesz isomorphism then X and Y are homeomorphic, and E and F are Riesz isomorphic. This generalizes the main results of [Z. Ercan, S. Önal, Banach-Stone theorem for Banach lattice valued continuous functions, Proc. Amer. Math. Soc. 135 (9) (2007) 2827-2829] and [X. Miao, C. Xinhe, H. Jiling, Banach-Stone theorems and Riesz algebras, J. Math. Anal. Appl. 313 (1) (2006) 177-183], and answers a conjecture in [Z. Ercan, S. Önal, Banach-Stone theorem for Banach lattice valued continuous functions, Proc. Amer. Math. Soc. 135 (9) (2007) 2827-2829].     

On K-groups of operator algebras on HD n spaces and QD n spaces

We obtain the K-groups of the operator ideals contained in the class of Riesz operators. And based on the results, we calculate the K-groups of the operator algebras on HD nspaces and QDn spaces.     

Some C*-algebras properties preserved by tracial approximation

We show that the Riesz interpolation property of the K ₀-monoid of C*-algebras in the class Ω is inherited by simple unital C*-algebras in the class TAΩ, and the property of being an admissible target algebras of finite type in the class of Ω is inherited by unital C*-algebras in the class TAΩ.