Automorphisms of Order Structures of Abelian Parts of Operator Algebras and Their Role in Quantum Theory

It is shown that any order isomorphism between the structures of unital associative JB subalgebras of JB algebras is given naturally by a partially linear Jordan isomorphism. The same holds for nonunital subalgebras and order isomorphisms preserving the unital subalgebra. Finally, we recover usual action of time evolution group on a von Neumann factor from group of automorphisms of the structure of Abelian subalgebras.     

Spectral Lattices

Spectral orthomorphisms between the spectral lattices of JBW algebras which preserve the scales extend to Jordan homomorphisms for a large class of algebras. Spectral lattice homomorphism is automatically a σ-lattice homomorphism. The range projection map is, up to a Jordan homomorphism, the only natural map from the spectral lattice onto the projection lattice. Continuity of the range projection determines finiteness of the algebra in Murray–von Neumann comparison theory.     

The Lattice and Simplex Structure of States on Pseudo Effect Algebras

We study states, measures, and signed measures on pseudo effect algebras with some version of the Riesz Decomposition Property (RDP). We show that the set of all Jordan signed measures is always an Abelian Dedekind complete ℓ-group. Therefore, the state space of a pseudo effect algebra with RDP is either empty or a nonempty Choquet simplex or even a Bauer simplex. This will allow to represent states on pseudo effect algebras by standard integrals.     

Closed and σ-Finite Measures on the Orthogonal Projections

We characterize the connection between closed and σ-finite measures on orthogonal projections of von Neumann algebras.     

Caratheodory and the Foundations of Thermodynamics and Statistical Physics

Constantin Caratheodory offered the first systematic and contradiction free formulation of thermodynamics on the basis of his mathematical work on Pfaff forms. Moreover, his work on measure theory provided the basis for later improved formulations of thermodynamics and physics of continua where extensive variables are measures and intensive variables are densities. Caratheodory was the first to see that measure theory and not topology is the natural tool to understand the difficulties (ergodicity, approach to equilibrium, irreversibility) in the Foundations of Statistical Physics. He gave a measure-theoretic proof of Poincaré's recurrence theorem in 1919. This work paved the way for Birkhoff to identify later ergodicity as metric transitivity and for Koopman and von Neumann to introduce spectral analysis of dynamical systems in Hilbert spaces. Mixing provided an explanation of the approach to equilibrium but not of irreversibility. The recent extension of spectral theory of dynamical systems to locally convex spaces, achieved by the Brussels–Austin groups, gives new nontrivial time asymmetric spectral decompositions for unstable and/or non-integrable systems. In this way irreversibility is resolved in a natural way.     

The role of type III factors in quantum field theory

One of von Neumann's motivations for developing the theory of operator algebras and his and Murray's 1936 classification of factors was the question of possible decompositions of quantum systems into independent parts. For quantum systems with a finite number of degrees of freedom the simplest possibility, i.e. factors of type I in the terminology of Murray and von Neumann, are perfectly adequate. In relativistic quantum field theory (RQFT), on the other hand, factors of type III occur naturally. The same holds true in quantum statistical mechanics of infinite systems. In this brief review some physical consequences of the type III property of the von Neumann algebras corresponding to localized observables in RQFT and their difference from the type I case will be discussed. The cumulative effort of many people over more than 30 years has established a remarkable uniqueness result: The local algebras in RQFT are generically isomorphic to the unique, hyperfinite type III, factor in Connes' classification of 1973. Specific theories are characterized by the net structure of the collection of these isomorphic algebras for different space-time regions, i.e. the way they are embedded into each other     

Classes of Invariant Subspaces for Some Operator Algebras

New results showing connections between structural properties of von Neumann algebras and order theoretic properties of structures of invariant subspaces given by them are proved. We show that for any properly infinite von Neumann algebra M there is an affiliated subspace \({\mathcal{L}} \) such that all important subspace classes living on \({\mathcal{L}} \) are different. Moreover, we show that \({\mathcal{L}} \) can be chosen such that the set of σ-additive measures on subspace classes of \({\mathcal{L}} \) are empty. We generalize measure theoretic criterion on completeness of inner product spaces to affiliated subspaces corresponding to Type I factor with finite dimensional commutant. We summarize hitherto known results in this area, discuss their importance for mathematical foundations of quantum theory, and outline perspectives of further research.     

Remarks on Effect Algebras

Erik M. Alfsen and Frederic W. Shultz had recently developed the characterisation of state spaces of operator algebras. It established full equivalence (in the mathematical sense) between the Heisenberg and the Schr?dinger picture, i.e. given a physical system we are able to construct its state space out of its observables as well as to construct algebra of observables from its state space. As an underlying mathematical structure they used the theory of duality of ordered linear spaces and obtained results are valid for various types of operator algebras (namely C ^*, von Neumann, JB and JBW algebras). Here, we show that the language they developed also admits a representation of an effect algebra.     

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.     

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.     

Field algebras do not leave field domains invariant

It is proved that von Neumann algebras associated to Op*-algebra (P, D) cannot leave the domainD ofP invariant if they are type I or type III factors or finite direct sums of such factors. Hence it follows that in quantum field theory global and local von Neumann field algebras in typical cases do not leave invariant the definition domain of Wightman fields.     

Continuity of the von Neumann Entropy

A general method for proving continuity of the von Neumann entropy on subsets of positive trace-class operators is considered. This makes it possible to re-derive the known conditions for continuity of the entropy in more general forms and to obtain several new conditions. The method is based on a particular approximation of the von Neumann entropy by an increasing sequence of concave continuous unitary invariant functions defined using decompositions into finite rank operators. The existence of this approximation is a corollary of a general property of the set of quantum states as a convex topological space called the strong stability property. This is considered in the first part of the paper.     

Unbounded generalizations of standard von Neumann algebras

The primary purpose of this paper is to study unbounded operator algebras called standard EW^#-algebras which are generalizations of standard von Neumann algebras to the unbounded case.The first part is an investigation of the weak, σ-weak, strong and σ-strong topologies on a standard EW^#-algebra. It is showed that the weak and σ-weak topologies on a standard EW^#-algebra coincide.The second part is a study of a locally convex topology called _∞L^ω₂-topology on a standard EW^#-algebra U. It is proved that U is a GB¹-algebra under the topology.The third part is an investigation of isomorphisms of standard EW^#-algebras. It is showed that if standard EW^#-algebras U and B are isomorphic then U and B are spatially isomorphic.     

De Morgan Property for Effect Algebras of von Neumann Algebras

It is shown that the unit interval of a von Neumann algebra is a Sum Brouwer–Zadeh algebra when equipped with another unary operation sending each element to the complement of its range projection. The main result of this Letter says that a von Neumann algebra is finite if and only if the corresponding Brouwer–Zadeh structure is de Morgan or, equivalently, if the range projection map preserves infima in the unit interval. This provides a new characterization of finiteness in the Murray–von Neumann structure theory of von Neumann algebras in terms of Brouwer–Zadeh structures.     

On the interplay of the centre and the state space in quantum logics

In some quantum system axiomatics (e.g. in the von Neumann formalism) the state space determines the centre. We show that in the logical-algebraic approach there is no kind of dependence (i.e. arbitrary two Boolean algebras may be the centres of two logics whose state spaces are identical).     

A Representation of Quantum Measurement in Nonassociative Algebras

Starting from an abstract setting for the Lüders-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., power-associativity or the sum postulate for observables) might turn out to be redundant then.     

Measure on the inductive limit of projection lattices

A probability measure on a nondecreasing net of lattices of orthogonal projections in von Neumann algebras is extended to a probability on the inductive limit of the lattices.     

A Representation of Quantum Measurement in Order-Unit Spaces

A certain generalization of the mathematical formalism of quantum mechanics beyond operator algebras is considered. The approach is based on the concept of conditional probability and the interpretation of the Lüders-von Neumann quantum measurement as a probability conditionalization rule. A major result shows that the operator algebras must be replaced by order-unit spaces with some specific properties in the generalized approach, and it is analyzed under which conditions these order-unit spaces become Jordan algebras. An application of this result provides a characterization of the projection lattices in operator algebras.     

A Hierarchy of Compatibility and Comeasurability Levels in Quantum Logics with Unique Conditional Probabilities

In the quantum mechanical Hilbert space formalism, the probabilisticinterpretation is a later ad-hoc add-on, more or less enforced by theexperimental evidence, but not motivated by the mathematical model itself. Amodel involving a clear probabilistic interpretation from the very beginningis provided by the quantum logics with unique conditional probabilities. Itincludes the projection lattices in von Neumann algebras and hereprobability conditionalization becomes identical with the state transitionof the Lüders - von Neumann measurement process. This motivates thedefinition of a hierarchy of five compatibility and comeasurability levelsin the abstract setting of the quantum logics with unique conditionalprobabilities. Their meanings are: the absence of quantum interference orinfluence, the existence of a joint distribution, simultaneous measurability, and the independence of the final state after two successive measurements from the sequential order of these two measurements. A further level means that two elements of the quantum logic (events) belong to the same Boolean subalgebra. In the general case, the five compatibility and comeasurability levels appear to differ, but they all coincide in the common Hilbert space formalism of quantum mechanics, in von Neumann algebras, and in some other cases.     

States on Operator Algebras and Axiomatic System of Quantum Theory

Recent development brings new results on the interplay of states on operator algebras and axiomatics of quantum mechanics. Neither hidden space in the sense of Kochen and Specker nor approximate hidden variables exist on von Neumann algebras. Tracial properties of states are connected with dispersions. The axioms on composite systems simplify to state extension properties.