Orthogonally additive functions on operator algebras

For a real or complex Hilbert spaceH, a general form of an even, orthogonally additive functional on B(H)is given.     

Positive linear maps on operator algebras

Given a family of completely positive maps, indexed by a group, from aC*-algebra into itself, we are concerned with its dilation to a group of *-automorphisms on a larger algebra. A Schwarz-type inequality forn-positive *-linear mappings from an involutive algebra into the bounded linear operators on a hilbert space is obtained. Strongly continuous one-parameter semigroups and groups onC*-algebras, which have certain positivity properties, are studied.     

Extension properties of states on operator algebras

We summarize and deepen some recent results concerning the extension problem for states on operator algebras and general quantum logics. In particular, we establish equivalence between the Gleason extension property, the Hahn-Banach extension property, and the universal state extension property of projection logics. Extensions of Jauch-Piron states are investigated.     

On topologization of unbounded operator algebras

The paper deals with the problem of whether or not for an Op¹-algebra $\text{A}$ the topologies τ$\text{D}$ and τ₀ or τ^($\text{D}$) and τ⁰ coincide on $\text{A}$. For certain Op¹-algebras which are dominated by a countably generated commutative subalgebra a necessary and sufficient condition is given. Some examples and related results are discussed.     

Induced modules for vertex operator algebras

For a vertex operator algebraV and a vertex operator subalgebraV which is invariant under an automorphismg ofV of finite order, we introduce ag-twisted induction functor from the category ofg-twistedV-modules to the category ofg-twistedV-modules. This functor satisfies the Frobenius reciprocity and transitivity. The results are illustrated withV being theg-invariants in simpleV orV beingg-rational.The first author was supported by NSF grant DMS-9303374 and a research grant from the Committee on Research, UC Santa Cruz.The second author was supported by NSF grant DMS-9401389.     

Derivations and automorphisms of operator algebras

The theorem that each derivation of aC*-algebra extends to an inner derivation of the weak-operator closure ( )^– of in each faithful representation of is proved in sketch and used to study the automorphism group of in its norm topology. It is proved that the connected component of the identity in this group contains the open ball of radius 2 with centerl and that each automorphism in extends to an inner automorphism of ( )^–.Research conducted with the partial support of the NSF and ONR.     

Probability operator of a harmonic oscillator

A unique linear rule of constructing quantum operators defined by the probability operator for coordinates and momenta, is considered. is assumed to be a normalized, positive definite operator, establishing a dynamical correspondence between the classical and quantum Poisson brackets. It is shown that such an operator exists in the case of a harmonic oscillator. The principal implications of the suggested rule of constructing the operators of physical quantities are determined, in comparison with the corresponding results of conventional quantum mechanics.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 89–93, October, 1982.     

Perturbations of flows on Banach spaces and operator algebras

For automorphism groups of operator algebras we show how properties of the difference _t – ' _t are reflected in relations between the generators , . Indeed for a von Neumann algebraM with separable predual we show that if _t – '_t 0.28 for smallt, then = 0(+)°^-1 where is an inner automorphism ofM and is a bounded derivation ofM. If the difference _t – ' _t =O(t) ast ; 0, then = + and if _t – ' _t 0.28 for allt then =. We prove analogous results for unitary groups on a Hilbert space andC ₀,C ₀ ^* groups on a Banach space.This paper subsumes an earlier work of the same title which appeared as a report from Z.I.F. der Universität BielefeldWith partial support of the U.S. National Science Foundation     

Analytic continuation of Moyal and operator algebras

We consider the analytic continuation of the commutator[q ^m, pⁿ for values of m,n positive integers. This is carried out both in the Moyal algebra and the operator algebra.     

Dissipative operators and cohomology of operator algebras

It is proved that an ultraweakly continuous completely dissipative operator on a W ^*-algebra A has a canonical form in terms of a completely positive map and a Hamiltonian provided that the cohomology groups of A with coefficients in a dual normal A-module are zero.     

Some operator algebras in nested Hilbert spaces

We prove the existence of a von Neumann algebra of operators and hence the existence of projections acting in any nested Hilbert space. Some other algebras of operators are studied. All those algebras are exhibited in a particular class of Nested Hilbert Spaces, namely sequence spaces.Aspirant au Fonds National de la Recherche Scientifique.     

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.     

Quantum measures and states on Jordan algebras

A problem of Mackey for von Neumann algebras has been settled by the conjunction of the early work of Gleason and the recent advances of Christensen and Yeadon. We show that Mackey's conjecture holds in much greater generality. LetA be a JBW-algebra and letL be the lattice of all projections inA. A quantum measure onL is a countably additive map,m, fromL into the real numbers. Our results imply thatm always has a unique extension to a bounded linear functional onA, provided thatA has no TypeI ₂ direct summand.     

Differential operator Lie algebras on the ring of Laurent polynomials

A class of differential operator Lie algebras on the unit circle is introduced and discussed. They are the natural generalizations of the Witt algebra and the Virasoro algebra. Among them are the higher-spin algebrasW ₁₊ andW which occur in the physics literature.     

On trace representation of linear functionals on unbounded operator algebras

In this paper we consider the following problem: Given a *-algebraA of unbounded operators, under what conditions is every strongly positive linear functionalf onA a trace functional, i.e. of the formf(a)=Trtta,aA, wheret is an appropriate positive nuclear operator. Further, the linear functionalsf onA which can be represented asf(a)=Trta (f andt not necessarily positive) are characterized by their continuity in a certain topology. Some applications (canonical commutation relations on the Schwartz space, integrable representations of enveloping algebras) are discussed.     

On the classification of simple vertex operator algebras

Inspired by a recent work of Frenkel-Zhu, we study a class of (pre-)vertex operator algebras (voa) associated to the self-dual Lie algebras. Based on a few elementary structural results we propose thatV, the category of Z₊-graded prevoasV in whichV[0] is one-dimensional, is a proper setting in which to study and classify simple objects. The categoryV is organized into what we call the minimalk ^th types. We introduce a functor —which we call the Frenkel-Lepowsky-Meurman functor—that attaches to each object inV a Lie algebra. This is a key idea which leads us to a (relative) classification of thesimple minimal first type. We then study the set of all Virasoro structures on a fixed minimal first typeV, and show that they are in turn classified by the orbits of the automorphism group Aut((V)) in cent((V)). Many new examples of voas are given. Finally, we introduce a generalized Kac-Casimir operator and give a simple proof of the irreducibility of the prolongation modules over the affine Lie algebras.     

Simple currents and extensions of vertex operator algebras

We consider how a vertex operator algebra can be extended to an abelian interwining algebra by a family of weak twisted modules which aresimple currents associated with semisimple weight one primary vectors. In the case that the extension is again a vertex operator algebra, the rationality of the extended algebra is discussed. These results are applied to affine Kac-Moody algebras in order to construct all the simple currents explicitly (except forE ₈) and to get various extensions of the vertex operator algebras associated with integrable representations.Supported by NSF grant DMS-9303374 and a research grant from the Committee on Research, UC Santa Cruz.Supported by NSF grant DMS-9401272 and a research grant from the Committee on Research, UC Santa Cruz.     

A remark on matrix product operator algebras,anyons and subfactors

Letters in Mathematical Physics - We show that the mathematical structures in a recent work of...     

Vertex operator representation of some quantum tori Lie algebras

We are defining the trigonometric Lie subalgebras in which are the natural generalization of the well known Sin-Lie algebra. The embedding formulas into are introduced. These algebras can be considered as some Lie algebras of quantum tori. An irreducible representation ofA, B series of trigonometric Lie algebras is constructed. Special cases of the trigonometric Lie factor algebras, which can be considered as a quantum (preserving Lie algebra structure) deformation of the Kac-Moody algebras are considered.