Jordan Operator Algebras Research Articles

Real Operator Algebras and Real Positive Maps

We present some foundations for a theory of real operator algebras and real Jordan operator algebras, and the various morphisms between these. A common theme is the ingredient of real positivity from papers of the first author with Read, Neal, Wang, and other coauthors, which we import to the real scalar setting here.

Commutativity in Jordan operator algebras

While Jordan algebras are commutative, operator commutativity will behave badly for a general non-associative Jordan algebra. When the product operators of two elements commute, the elements are said to operator commute. In some Jordan algebras operator commutation can be badly behaved, for instance having elements a and b operator commute, while a2 and b do not operator commute. In this paper we study JB-algebras, real Jordan algebras which are also Banach spaces in a compatible manner. These include for instance the spaces of self-adjoint elements of unital C⁎-algebras. We show that elements a and b in a JB-algebra operator commute if and only if they generate an associative subalgebra of mutually operator commuting elements, and hence operator commutativity in JB-algebras is as well-behaved as it can be. Letting Ua denote the quadratic operator of a, we also show that positive a and b operator commute if and only if Uab2=Uba2. We use this result to conclude that the unit interval of a JB-algebra is a sequential effect algebra as defined by Gudder and Greechie.

Jordan operator algebras revisited

AbstractJordan operator algebras are norm‐closed spaces of operators on a Hilbert space with for all . In two recent papers by the authors and Neal, a theory for these spaces was developed. It was shown there that much of the theory of associative operator algebras, in particular, surprisingly much of the associative theory from several recent papers of the first author and coauthors, generalizes to Jordan operator algebras. In the present paper we complete this task, giving several results which generalize the associative case in these papers, relating to unitizations, real positivity, hereditary subalgebras, and a couple of other topics. We also solve one of the three open problems stated at the end of our earlier joint paper on Jordan operator algebras.

Noncommutative topology and Jordan operator algebras

AbstractJordan operator algebras are norm‐closed spaces of operators on a Hilbert space with for all . We study noncommutative topology, noncommutative peak sets and peak interpolation, and hereditary subalgebras of Jordan operator algebras. We show that Jordan operator algebras present perhaps the most general setting for a “full” noncommutative topology in the ‐algebraic sense of Akemann, L. G. Brown, Pedersen, etc, and as modified for not necessarily selfadjoint algebras by the authors with Read, Hay and other coauthors. Our breakthrough relies in part on establishing several strong variants of ‐algebraic results of Brown relating to hereditary subalgebras, proximinality, deeper facts about for a left ideal L in a ‐algebra, noncommutative Urysohn lemmas, etc. We also prove several other approximation results in ‐algebras and various subspaces of ‐algebras, related to open and closed projections and technical ‐algebraic results of Brown.

Jordan operator algebras: basic theory

Jordan operator algebras are norm-closed spaces of operators on a Hilbert space which are closed under the Jordan product. The discovery of the present paper is that there exists a huge and tractable theory of possibly nonselfadjoint Jordan operator algebras; they are far more similar to associative operator algebras than was suspected. We initiate the theory of such algebras.

STABILITY OF (α, β, γ)-DERIVATIONS ON LIE C*-ALGEBRAS

Petr Novotný and Jiřĺ Hrivnák [14] investigated and generalized the concept of Lie derivations via certain complex parameters and obtained various Lie and Jordan operator algebras as well as two one-parametric sets of linear operators. Moreover, they established the structure and properties of (α, β, γ)-derivations of Lie algebras. We say a functional equation (ξ) is stable if any function g satisfying the equation (ξ) approximately is near to true solution of (ξ). In the present paper, we investigate the stability of (α, β, γ)-derivations on Lie C*-algebras associated with the following functional equation [Formula: see text]

A Representation of Quantum Measurement in Nonassociative Algebras

Starting from an abstract setting for the Lueders - von Neumann quantum measurement process and its interpretation as a probability conditionalization rule in a non-Boolean event structure, the author derived a certain generalization of operator algebras in a preceding paper. This is an order-unit space with some specific properties. It becomes a Jordan operator algebra under a certain set of additional conditions, but does not own a multiplication operation in the most general case. A major objective of the present paper is the search for such examples of the structure mentioned above that do not stem from Jordan operator algebras; first natural candidates are matrix algebras over the octonions and other nonassociative rings. Therefore, the case when a nonassociative commutative multiplication exists is studied without assuming that it satisfies the Jordan condition. The characteristics of the resulting algebra are analyzed. This includes the uniqueness of the spectral resolution as well as a criterion for its existence, subalgebras that are Jordan algebras, associative subalgebras, and more different levels of compatibility than occurring in standard quantum mechanics. However, the paper cannot provide the desired example, but contribute to the search by the identification of some typical differences between the potential examples and the Jordan operator algebras and by negative results concerning some first natural candidates. The possibility that no such example exists cannot be ruled out. However, this would result in an unexpected new characterization of Jordan operator algebras, which would have a significant impact on quantum axiomatics since some customary axioms (e.g., powerassociativity or the sum postulate for observables) might turn out to be redundant then.

Additivity of Jordan maps on standard Jordan operator algebras

Let A be a standard Jordan operator algebra on a Hilbert space of dimension > 1 and B be an arbitrary Jordan algebra. In this note, we prove that if a bijection ϕ : A → B satisfies ϕ ( a ∘ b ) = ϕ ( a ) ∘ ϕ ( b ) for all a , b ∈ A , where ∘ is the special Jordan product on A, then ϕ is additive.

On [formula omitted]-derivations of Lie algebras and corresponding invariant functions

We consider finite dimensional complex Lie algebras. We generalize the concept of Lie derivations via certain complex parameters and obtain various Lie and Jordan operator algebras as well as two one-parametric sets of linear operators. Using these parametric sets, we introduce complex functions with a fundamental property — invariance under Lie isomorphisms. One of these basis-independent functions represents a complete set of invariant(s) for three-dimensional Lie algebras. We present also its application to physically motivated examples in dimension 8.

Different Types of Conditional Expectation and the Lüders—von Neumann Quantum Measurement

In operator algebra theory, a conditional expectation is usually assumed to be a projection map onto a sub-algebra. In the paper, a further type of conditional expectation and an extension of the Lüders—von Neumann measurement to observables with continuous spectra are considered; both are defined for a single operator and become a projection map only if they exist for all operators. Criteria for the existence of the different types of conditional expectation and of the extension of the Lüders—von Neumann measurement are presented, and the question whether they coincide is studied. All this is done in the general framework of Jordan operator algebras. The examples considered include the type I and type II operator algebras, the standard Hilbert space model of quantum mechanics, and a no-go result concerning the conditional expectation of observables that satisfy the canonical commutator relation.

Conditional expectations associated with quantum states

An extension of the conditional expectations (those under a given subalgebra of events and not the simple ones under a single event) from the classical to the quantum case is presented. In the classical case, the conditional expectations always exist; in the quantum case, however, they exist only if a certain weak compatibility criterion is satisfied. This compatibility criterion was introduced among others in a recent paper by the author. Then, state-independent conditional expectations and quantum Markov processes are studied. A classical Markov process is a probability measure, together with a system of random variables, satisfying the Markov property and can equivalently be described by a system of Markovian kernels (often forming a semigroup). This equivalence is partly extended to quantum probabilities. It is shown that a dynamical (semi)group can be derived from a given system of quantum observables satisfying the Markov property, and the group generators are studied. The results are presented in the framework of Jordan operator algebras, and a very general type of observables (including the usual real-valued observables or self-adjoint operators) is considered.

Composite systems and the role of the complex numbers in quantum mechanics

An axiomatic approach to the mathematical formalism of quantum mechanics, based upon a certain concept of conditional probability, has been proposed in two recent papers by the author. It leads to Jordan operator algebras and thus comes rather close to the standard Hilbert space model of quantum mechanics, but still includes the so-called exceptional Jordan algebras, for which a Hilbert space representation does not exist. This approach is now extended by defining a mathematical model of composite systems. Such a model is required for the study of the joint distribution of two quantum observables. A very general type of observables (not only the real-valued observables corresponding to the self-adjoint operators) is considered. The joint distribution is defined, using the concept of conditional probability, and exhibits a certain dependence on the succession of the observations which is different from the classical case and unknown so far in quantum mechanics. Finally, it turns out that, at least in the finite-dimensional case, a really satisfying model of the composite system exists only if each single system is modeled by a complex Jordan matrix algebra (or a direct sum), and the model then becomes the tensor product. This provides some reasoning why the exceptional Jordan algebras can be ruled out, why quantum mechanics needs the complex numbers and the complex Hilbert space, and why the tensor product is the right choice for the model of a composite system.

Why Do the Quantum Observables Form a Jordan Operator Algebra?

The Jordan algebra structure of the bounded real quantum observables was recognized already in the early days of quantum mechanics. While there are plausible reasons for most parts of this structure, the existence of the distributive nonassociative multiplication operation is hard to justify from a physical or statistical point of view. Considering the non-Boolean extension of classical probabilities, presented in a recent paper, it is shown in this paper that such a multiplication operation can be derived from certain properties of the conditional probabilities and the observables, i.e., from postulates with a clear statistical interpretation. The well-known close relation between Jordan operator algebras and C*-algebras then provides the connection to the quantum-mechanical Hilbert space formalism, thus resulting in a novel axiomatic approach to general quantum mechanics that includes the types II and III von Neumann algebras.

Non-Boolean probabilities and quantum measurement

A non-Boolean extension of the classical probability model is proposed. The non-Boolean probabilities reproduce typical quantum phenomena. The proposed model is more general and more abstract, but easier to interpret, than the quantum mechanical Hilbert space formalism and exhibits a particular phenomenon (state-independent conditional probabilities) which may provide new opportunities for an understanding of the quantum measurement process. Examples of the proposed model are provided, using Jordan operator algebras.

Continuity and linear extensions of quantum measures on Jordan operator algebras.

Let M be a W * -algebra on a JBW-algebra. A (finitely additive) quantum measure on M is a non-negative real valued function μ on the projections on M which satisfies the property μ(p 1 +...p n )=μ(p 1 )+...+μ(p n ), μ(1)=1 whenever p 1 ,...,p n are orthogonal projections of M

A new proof of the existence of traces on Jordan operator algebras and real von Neumann algebras

In this paper we show that a JW-algebra is modular if and only if it admits a separating family of finite normal traces. The proof does not follow the von Neumann algebras case but is based on the finiteness of the enveloping von Neumann algebra for a reversible modular JW-algebra. For a locally modular JW-algebra the existence of a faithful semifinite normal trace is proved. Similar results are obtained for respectively finite and semifinite real von Neumann algebras.

Jordan operator algebras

A survey of recent results in classification of JW-algebras (weakly closed Jordan algebras of self-adjoint operators in the Hilbert space) is given along with connections of JW-algebras with their enveloping W*-algebras. It is shown how these results are applied in the proofs of analogs of many important results in the theory of W*-algebras.

H. Hanche-Olsen and E. Stormer Jordan operator algebras (Monographs and Studies in Mathematics, Vol. 21, Pitman, 1984), 183 pp. £27.50.

An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Certain Jordan operator algebras and double commutant theorems

Two operations on sets of bounded operators on Hilbert space are studied: One produces Jordan algebras and the other, its inverse in a certain sense, produces self-adjoint linear spaces. The question of which Jordan algebras arise from the first of these operations is investigated. It is shown that, in many cases, R has this property if and only if the complex associative algebra R + iR is equal to its double commutant. Thus, a by-product of the study is another approach to double commutant theorems.