Interval Effect Algebras Research Articles

Spectral resolutions in effect algebras

Effect algebras were introduced as an abstract algebraic model for Hilbert space effects representing quantum mechanical measurements. We study additional structures on an effect algebraEthat enable us to define spectrality and spectral resolutions for elements ofEakin to those of self-adjoint operators. These structures, called compression bases, are special families of maps onE, analogous to the set of compressions on operator algebras, order unit spaces or unital abelian groups. Elements of a compression base are in one-to-one correspondence with certain elements ofE, called projections. An effect algebra is called spectral if it has a distinguished compression base with two special properties: the projection cover property (i.e., for every elementainEthere is a smallest projection majorizinga), and the so-called b-comparability property, which is an analogue of general comparability in operator algebras or unital abelian groups. It is shown that in a spectral archimedean effect algebraE, everya∈Eadmits a unique rational spectral resolution and its properties are studied. If in additionEpossesses a separating set of states, then every elementa∈Eis determined by its spectral resolution. It is also proved that for some types of interval effect algebras (with RDP, archimedean divisible), spectrality ofEis equivalent to spectrality of its universal group and the corresponding rational spectral resolutions are the same. In particular, for convex archimedean effect algebras, spectral resolutions inEare in agreement with spectral resolutions in the corresponding order unit space.

Effect Algebras of Positive Linear Operators Densely Defined on Hilbert Spaces

We show that the set of all positive linear operators densely defined in an infinite-dimensional complex Hilbert space can be equipped with partial sum of operators making it a generalized effect algebra. This sum coincides with the usual sum of two operators whenever it exists. Moreover, blocks of this generalized effect algebra are proper sub-generalized effect algebras. All intervals in this generalized effect algebra become effect algebras which are Archimedean, convex, interval effect algebras, for which the set of vector states is order determining. Further, these interval operator effect algebras possess faithful states.

Extensions of Witness Mappings

We deal with the problem of coexistence in interval effect algebras using the notion of a witness mapping. Suppose that we are given an interval effect algebra $E$, a coexistent subset $S$ of $E$, a witness mapping $\beta$ for $S$, and an element $t\in E\setminus S$. We study the question whether there is a witness mapping $\beta_t$ for $S\cup\{t\}$ such that $\beta_t$ is an extension of $\beta$. In the main result, we prove that such an extension exists if and only if there is a mapping $e_t$ from finite subsets of $S$ to $E$ satisfying certain conditions. The main result is then applied several times to prove claims of the type "If $t$ has a such-and-such relationship to $S$ and $\beta$, then $\beta_t$ exists".

Extensions of Effect Algebra Operations

We study the set of all positive linear operators densely defined in an infinite-dimensional complex Hilbert space. We equip this set with various effect algebraic operations making it a generalized effect algebra. Further, sub-generalized effect algebras and interval effect algebras with respect of these operations are investigated.

Coexistence in interval effect algebras

Motivated by the notion of coexistence of effect-valued observables, we give a characterization of coexistent subsets of interval effect algebras.

Every state on interval effect algebra is integral

We show that every state on an interval effect algebra is an integral through some regular Borel probability measure defined on the Borel σ-algebra of a compact Hausdorff simplex. This is true for every effect algebra satisfying Riesz decomposition property or for every many valued (MV)-algebra. In addition, we show that each state on an effect subalgebra of an interval effect algebra E can be extended to a state on E. Our method represents also every state on the set of effect operators of a Hilbert space as an integral.

Interval and Scale Effect Algebras

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Chain tensor products and interval effect algebras

Weak and strongn-doublings (n∈N) are defined for an effect algebraP and the concept of a normal interval algebra is introduced. It is shown that the following statements are equivalent: (1) There is a morphism fromP into an interval algebra. (2)P admits a tensor product with every finite chain. (3)P has a weakn-doubling for everyn∈N. Moreover, the following are equivalent: (4)P is a normal interval algebra. (5)P admits a strong tensor product with every chain of length 2n,n∈N. (6)P has a strongn-doubling for everyn∈N. Finally, it is shown that ifP possesses an order-determining set of states, thenP is a normal interval algebra.

Quotients of interval effect algebras

Nearly every orthostructure that has been proposed as a model for a logic of propositions affiliated with a physical system can be represented as an interval effect algebra; that is, as the partial algebra under addition of an interval from zero to an order unit in a partially ordered Abelian group. If the system is in a state that precludes certain elements of such an interval, an appropriate quotient interval algebra can be constructed by factoring out the order-convex subgroup generated by the precluded elements. In this paper we launch a study of the resulting quotient effect algebras.

Phi-symmetric effect algebras

The notion of a Sasaki projectionon an orthomodular lattice is generalized to a mapping Φ: E × E → E, where E is an effect algebra. If E is lattice ordered and Φ is symmetric, then E is called a Φ-symmetric effect algebra.This paper launches a study of such effect algebras. In particular, it is shown that every interval effect algebra with a lattice-ordered ambient group is Φ-symmetric, and its group is the one constructed by Ravindran in his proof that every effect algebra that has the Riesz decomposition property is an interval algebra. It is shown that the doubling construction introduced in the paper is connected to the conditional event algebrasof Goodman, Nguyen, and Walker.