Automorphisms of quantum logics

Automorphisms of quantum logics are studied. If a quantum logic, i.e. an orthomodular complete lattice of propositions concerning a physical system, is represented as the lattice of all projections in a von Neumann algebra, then each automorphism of the logic can be represented as a Jordan automorphism in the algebra. Groups of transformations of a physical system are represented as groups of ¹-automorphisms in a von Neumann algebra, provided certain continuity conditions are fulfilled.     

Clustering for algebraicK-systems

We prove that for a von Neumann algebra that is an algebraicK system with respect to some automorphism, the invariant state isK-clustering andr-clustering. Further, we study by using examples how far the von Neumann algebra inherits theK property from the underlyingC ^* algebra.     

States and automorphism groups of operator algebras

Suppose that a group of automorphisms of a von Neumann algebraM, fixes the center elementwise. We show that if this group commutes with the modular (KMS) automorphism group associated with a normal faithful state onM, then this state is left invariant by the group of automorphisms. As a result we obtain a “noncommutative” ergodic theorem. The discrete spectrum of an abelian unitary group acting as automorphisms ofM is completely characterized by elements inM. We discuss the KMS condition on the CAR algebra with respect to quasi-free automorphisms and gauge invariant generalized free states. We also obtain a necessary and sufficient condition for the CAR algebra and a quasi-free automorphism group to be η-abelian.     

On invariant states and the commutant of a group of quasi-free automorphisms of the CAR algebra,II

We improve Theorem 3 of a previous paper [3] as follows: If M is a von Neumann algebra on the one-particle space which does not contain any finite type I factor direct summands, an automorphism of the CAR algebra which carries the set of quage-invariant quasi-free states Φ_A with A in M onto itself is quasi-free.     

Construction of a left Hilbert algebra with respect to a Minkowski form

We construct a left Hilbert algebra with respect to a Minkowski form and generalize the theorem that every von Neumann algebra is isomorphic to the left von Neumann algebra of a left Hilbert algebra.     

Inner automorphisms of von Neumann algebras

It is first shown that a *-automorphism of a factor is inner if and only if it is asymptotically equal to the identity automorphism. Then it is shown that a periodic *-automorphism of a von Neumann algebra is inner if and only if its fixed point algebra is a normal subalgebra of .     

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.     

Measures on Classes of Subspaces Affiliated with a von Neumann Algebra

Let ℳ be a von Neumann algebra acting on a Hilbert space H and let S be a dense lineal in H that is affiliated with a von Neumann algebra ℳ. The “topological” definition of measures on the classes of orthoclosed and splitting subspaces of S affiliated with a von Neumann algebra ℳ is given and results on the relationships of these measures to measures on orthoprojections of the algebra ℳ are presented.     

Golden-Thompson and Peierls-Bogolubov inequalities for a general von Neumann algebra

Some inequalities for a general von Neumann algebra, which reduces to Golden-Thomspon and Peierls-Bogolubov inequalities when the von Neumann algebra has a trace, are proved.     

On locally normal states in quantum statistical mechanics

A sufficient condition is given in order that a von Neumann algebra with cyclic vector is quasi-standard. With the help of this result it is proved that a locally normal state with a cyclic and separating vector in the representation space gives rise to a quasi-standard von Neumann algebra. Furthermore it is proved that the representation space determined by a locally normal state in the G.N.S. construction is separable.     

Type-Decomposition of a Synaptic Algebra

A synaptic algebra is a generalization of the self-adjoint part of a von Neumann algebra. In this article we extend to synaptic algebras the type-I/II/III decomposition of von Neumann algebras, AW^?-algebras, and JW-algebras.     

Entropy of Quantum States

Given the algebra of observables of a quantum system subject to selection rules, a state can be represented by different density matrices. As a result, different von Neumann entropies can be associated with the same state. Motivated by a minimality property of the von Neumann entropy of a density matrix with respect to its possible decompositions into pure states, we give a purely algebraic definition of entropy for states of an algebra of observables, thus solving the above ambiguity. The entropy so-defined satisfies all the desirable thermodynamic properties and reduces to the von Neumann entropy in the quantum mechanical case. Moreover, it can be shown to be equal to the von Neumann entropy of the unique representative density matrix belonging to the operator algebra of a multiplicity-free Hilbert-space representation.     

Modular structure of the local algebras associated with the free massless scalar field theory

The modular structure of the von Neumann algebra of local observables associated with a double cone in the vacuum representation of the free massless scalar field theory of any number of dimensions is described. The modular automorphism group is induced by the unitary implementation of a family of generalized fractional linear transformations on Minkowski space and is a subgroup of the conformal group. The modular conjugation operator is the anti-unitary implementation of a product of time reversal and relativistic ray inversion. The group generated by the modular conjugation operators for the local algebras associated with the family of double cone regions is the group of proper conformal transformations. A theorem is presented asserting the unitary equivalence of local algebras associated with lightcones, double cones, and wedge regions. For the double cone algebras, this provides an explicit realization of spacelike duality and establishes the known typeIII ₁ factor property. It is shown that the timelike duality property of the lightcone algebras does not hold for the double cone algebras. A different definition of the von Neumann algebras associated with a region is introduced which agrees with the standard one for a lightcone or a double cone region but which allows the timelike duality property for the double cone algebras. In the case of one spatial dimension, the standard local algebras associated with the double cone regions satisfy both spacelike and timelike duality.Supported by the National Science Foundation under Grant No. PHY-79-23251Supported in part by C. N. R.     

Random Operators and Crossed Products

This article is concerned with crossed products and their applications to random operators. We study the von Neumann algebra of a dynamical system using the underlying Hilbert algebra structure. This gives a particularly easy way to introduce a trace on this von Neumann algebra. We review several formulas for this trace, show how it comes as an application of Connes" noncommutative integration theory and discuss Shubin"s trace formula. We then restrict ourselves to the case of an action of a group on a group and include new proofs for some theorems of Bellissard and Testard on an analogue of the classical Plancherel theorem. We show that the integrated density of states is a spectral measure in the periodic case, thereby generalizing a result of Kaminker and Xia. Finally, we discuss duality results and apply a method of Gordon et al. to establish a duality result for crossed products by Z.     

Subspace Structures in Inner Product Spaces and von Neumann Algebras

We study subspaces of inner product spaces that are invariant with respect to a given von Neumann algebra. The interplay between order properties of the poset of affiliated subspaces and the structure of a von Neumann algebra is investigated. We extend results on nonexistence of measures on incomplete structures to invariant subspaces. Results on inner product spaces as well as on the structure of affiliated subspaces are reviewed.     

Representations of von Neumann Algebras and Ultraproducts

International Journal of Theoretical Physics - We will introduce the concept of ergodicity of states with respect to some group of transformations on a von Neumann algebra and its properties are...     

States on projection logics of von Neumann algebras

We summarize recent results concerning states on projection lattices of von Neumann algebras. In particular, we present an analysis of the Jauch-Piron property in the von Neumann algebra setting.     

Spectral resolution in a Rickart comgroup

A comgroup is a compressible group with the general comparability property. A comgroupwith the Rickart projection property is called a Rickart comgroup. We show that each element of a Rickart comgroup has a rational spectral resolution and a nonempty closed and bounded (real) spectrum. The rational spectral resolution and the spectrum are shown to have many of the properties of the spectral resolution and spectrum of a self-adjoint operator on a Hilbert space. Examples of Rickart comgroups include the additive group of self-adjoint elements in a von Neumann algebra and the Mundici group of a Heyting MV algebra.     

A Riemann–Roch Theorem¶for One-Dimensional Complex Groupoids

We consider a smooth groupoid of the form Σ⋊Γ, where Σ is a Riemann surface and Γ a discrete pseudogroup acting on Σ by local conformal diffeomorphisms. After defining a K-cycle on the crossed product C ₀(Σ)⋊Γ generalising the classical Dolbeault complex, we compute its Chern character in cyclic cohomology, using the index theorem of Connes and Moscovici. This involves in particular a generalisation of the Euler class constructed from the modular automorphism group of the von Neumann algebra L ^∞(Σ)⋊Γ. Received: 1 February 2000 / Accepted: 3 December 2000     

A duality theorem for the automorphism group of a covariant system

In this paper we examine the covariant representation theory of a covariant system (A, G) introduced by Doplicher, Kastler and Robinson. (A is aC*-algebra andG is a locally compact group of automorphisms ofA.) We define the concept of left tensor product of two covariant representations. Loosely stated, our duality theorem says thatG is canonically isomorphic to the set of bounded operator valued maps on the set of covariant representations of the covariant system (A, G) which preserve direct sums, unitary equivalence and left tensor products. We further show that the enveloping von Neumann algebraA(A, G) of the covariant system (A, G) admits a (not necessarily injective) comultiplicationd such that (A(A, G),d) is a Hopf von Neumann algebra. The intrinsic group of this Hopf von Neumann algebra is canonically isomorphic and (strong operator topology) homeomorphic toG.