Dissipations on von Neumann algebras

It is shown that for certain classes of Euclidean fermion-boson systems on a lattice vacuum expectation values of scalar fields increase if a Yukawa-interaction is turned on. Applicability and possible extensions of this result in the framework of constructive quantum-field-theory are discussed.Supported by Studienstiftung des deutschen Volkes     

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 .     

Perturbation theory of von Neumann entropy

In quantum information theory, von Neumann entropy plays an important role; it is related to quantum channel capacities. Only for a few states can one obtain their entropies. In a continuous variable system, numeric evaluation of entropy is not easy due to infinite dimensions. We develop the perturbation theory for systematically calculating von Neumann entropy of a non-degenerate system as well as a degenerate system.     

Dynamical approximation entropies and topological entropy in operator algebras

Dynamical entropy invariants, based on a general approximation approach are introduced for C^*-and W^*-algebra automorphisms. This includes a noncommutative extension of topological entropy.Supported in part by Grant No. DMS 9204174 from the National Science Foundation.     

Conditional expectation and commutativity in von Neumann algebras

LetH be a Hilbert space,A the von Neumann algebra of all bounded operators onH,B a von Neumann subalgebra ofA, andw a bounded linear functional onA. The functionalw is said to commute withBA ifw(AB)=w(BA) for allAA. It is shown that the mapBw (BAB) is a complex measure on the orthocomplemented partially ordered set of all orthogonal projections inB for everyAA if and only ifw commutes with all members ofB. For anyAA, the conditional expectation ofA with respect toB andw is defined and it is shown that this expectation exists for an Abelian separableB ifw commutes with all members ofB. Using Gleason's theorem it is shown thatw commutes withB if and only if the density operator ofw commutes withB.     

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.     

Inequalities for traces on von Neumann algebras

A number of useful inequalities, which are known for the trace on a separable Hilbert space, are extended to traces on von Neumann algebras. In particular, we prove the Golden rule, Hölder inequality, and some convexity statements.Battelle Fellow, 1970–1971.     

Dynamical entropy of quasi-local algebras in quantum statistical mechanics

We study the dynamical entropy in the sense of Connes, Narnhofer, and Thirring of automorphisms on quasi-local algebras in quantum statistical mechanics. We extend their Kolmogorov-Sinai type theorem for AF-algebras to quasi-local algebras which are not necessarily AF-algebras.Work supported in part by Korean Science Foundation and the Basic Science Research Program, Ministry of Education, 1991     

Symmetries and half-sided modular inclusions of von Neumann algebras

LetN, be a von Neumann algebras on a Hilbert space , a common cyclic and separating vector. Assume to be cyclic and also separating forN . Denote by , _N , _N the modular operators to (, ), (N, ), resp (N , ). Assume now ^-it N ^it N for allt 0. (Such type of inclusions ((N U, ) , ) are called half-sided modular.) Then the modular groups ^it , _N ^ir , _N ^is ,t, r, s generate a unitary representation of the group S1(2, )/Z ₂ of positive energy.Another result is related to two half-sided modular inclusions (₁ , ) and (₂ , ). Under proper conditions the three modular groups ^it , ₁ ^ir , ₂ ^is ,t, r, s generate the three-dimensional subgroup of O(2, 1) of two commuting translations and the Lorentz transformation.Partly supported by the DFG, SFB 288 Differentialgeometrie und Quantenphysik.     

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.     

Extension of covariant POV-measures in von Neumann algebras

We discuss the extension of POV-measures transforming covariantly with respect to automorphic group representations on von Neumann algebras.     

Unitary implementation of automorphism groups on von Neumann algebras

A necessary and sufficient continuity condition is obtained in order that a topological group of automorphisms of a semi-finite von Neumann algebra in standard form is unitarily implemented. The methods used are extended to the study of unitary implementation for a general von Neumann algebra of those automorphism groups that commute with the one-parameter modular automorphism group.This research was partially supported by the National Science Foundation.     

A new axiomatic characterization of the von Neumann entropy

We lay down a series of postulates which we consider indespensable for any measure of the intrinsic dispersion of quantum states. Then we show that the v. Neumann entropy S(W)=–kTr(WlnW) is uniquely determined by these postulates up to a scaling factor. Thus the v. Neumann entropy can be characterized as the only reasonable measure of the intrinsic dispersion of quantum states. Our postulates have no direct bearing on phenomenological thermodynamics or information theory.     

Frobenius theory for positive maps of von Neumann algebras

Frobenius theory about the cyclic structure of eigenvalues of irreducible non negative matrices is extended to the case of positive linear maps of von Neumann algebras. Semigroups of such maps and ergodic properties are also considered.     

Calculation of von Neumann entropy for hydrogen and positronium negative ions

In the present work, we carry out calculations of von Neumann entropies and linear entropies for the hydrogen negative ion and the positronium negative ion. We concentrate on the spatial (electron–electron orbital) entanglement in these ions by using highly correlated Hylleraas functions to represent their ground states, and to take care of correlation effects. We apply the Schmidt decomposition method on the partial-wave expanded two-electron wave functions, and from which the one-particle reduced density matrix can be obtained, leading to the quantifications of linear entropy and von Neumann entropy in the H⁻ and Ps⁻ ions.     

Remarks on spectra of modular operators of von Neumann algebras

It is shown that if is an invariant state of an asymptotically abelianC* algebra , then the spectrum of modular operator for is contained in the spectrum of any other modular operator for the von Neumann algebra .It is also shown that a modular operator can not have an isolated spectrum with a finite multiplicity at 1 unless the associated Hilbert space is of finite dimension. It is further shown that if a modular operator has an isolated spectrum with a finite multiplicity atx 1, then the von Neumann algebra is a direct sum of ₁ and ₂ where ₁ is represented on a finite dimensional Hilbert space and the modular operator for ₂ does not have its spectrum atx.Applications to Connes invariant are given.On leave from Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan.     

Types of von Neumann algebras associated with extremal invariant states

A globalized version of the following is proved. Let be a factor acting on a Hilbert space ,G a group of unitary operators on inducing automorphisms of ,x a vector separating and cyclic for which is up to a scalar multiple the unique vector invariant under the unitaries inG. Then either is of type III or _x is a trace of . The theorem is then applied to study the representations due to invariant factors state of asymptotically abelianC*-algebras, and to show that in quantum field theory certain regions in the Minkowski space give type III factors.     

Conditional expectations on von Neumann algebras: A new approach

In the framework of non-commutative probability theory on von Neumann algebras the concept of conditional expectation is redefined in such a way that it exists relative to arbitrary a priori distributions and subalgebras. We derive explicit expressions for the expectations relative to Abelian subalgebras and establish their connection with the concept of coarse-graining.     

Replacing energy by von Neumann entropy in quantum phase transitions

We propose that quantum phase transitions are generally accompanied by non-analyticities of the von Neumann (entanglement) entropy. In particular, the entropy is non-analytic at the Anderson transition, where it exhibits unusual fractal scaling. We also examine two dissipative quantum systems of considerable interest to the study of decoherence and find that non-analyticities occur if and only if the system undergoes a quantum phase transition.