Showing papers on "Operator algebra published in 1975"

Hereditary and intermediate reflexivity of w*-algebras

Abstract: An operator R is reflexive if every operator which leaves invariant all R-invariant subspaces belongs to R. The notion of reflexivity can be extended to linear spaces of operators. An operator algebra is said to be hereditarily reflexive if all its weakly closed subspaces are reflexive. This article presents a criterion for the hereditary reflexivity of a W*-algebra, and also examines the more general problem of conditions for the intermediate reflexivity of a pair of W*-algebras. A number of necessary conditions and sufficient conditions for intermediate reflexivity are also obtained.Bibliography: 20 titles.

Linear Response Theory and the KMS Condition

Abstract: The response, relaxation and correlation functions are defined for any vector state e of a von Neumann algebra\(\mathfrak{M}\), acting on a Hilbert space ℋ, satisfying the KMS-condition. An operator representation of these functions is given on a particular Hilbert space .

Integral representations for Schwinger functionals and the moment problem over nuclear spaces

Abstract: It is shown that a continuous positive linear functional on a commutative nuclear *-algebra has an integral decomposition into characters if and only if the functional is strongly positive, i.e. positive on all positive polynomials. When applied to the symmetric tensor algebra over a nuclear test function space this gives a necessary and sufficient condition for the Schwinger functions of Euclidean quantum field theory to be the moments of a continuous cylinder measure on the dual space. Another application is to the problem of decomposing a Wightman functional into states having the cluster property.

A generalization of the classical moment problem on $^*$-algebras with applications to relativistic quantum theory. II

Abstract: A (non-commutative) generalization of the classical moment problem is formulated on arbitrary *-algebras with units. This is used to produce aC*-algebra associated with the space of test functions for quantum fields. ThisC*-algebra plays a role in theories of bounded localized observables in Hilbert space which is similar to that of the space of test functions in quantum field theories (namely it is represented in Hilbert space). The case of local quantum fields which satisfy a slight generalization of the growth condition is investigated.

On the Algebra of Field Operators. The Weak Commutant and Integral Decompositions of States

Abstract: It is shown that every continuous representation of a nuclear and separable *-algebra on a separable Hilbert space has a certain integral decomposition into representations with a trivial weak commutant. This result is used to obtain a decomposition of Wightman functionals into extremal states.

Global dimension of differential operator rings. II

Abstract: This paper is concerned with finding the global homological dimension of the ring of differential operators R[0] over a differential ring R with a single derivation. Examples are constructed to show that R[O] may have finite dimension even when R has infinite dimension. For a commutative noetherian differential algebra R over the rationals, with finite global dimension n, it is shown that the global dimension of R[0] is the supremum of n and one plus the projective dimensions of the modules R/P, where P ranges over all prime differential ideals of R. One application derives the global dimension of the Weyl algebra over a commutative noetherian ring S of finite global dimension, where S either is an algebra over the rationals or else has positive characteristic.

Representations of a local current algebra: Their dynamical determination

Abstract: Local currents are used to describe nonrelativistic many‐body quantum mechanics in the thermodynamic limit. The problem of determining a representation of the local currents corresponding to a given Hamiltonian is studied. We formulate the dynamics in such a way that one solves simultaneously for the ground state and the representation of the local currents. This leads to two coupled functional equations relating the generating functional to a functional which describes the ground state. Together these functionals determine a representation of the local currents in which the Hamiltonian is a well‐defined operator. The functional equations are equivalent to a set of integro–differential equations for expansion coefficients of the two functionals.

On commutative normal *-derivations

Abstract: The notion of commutative derivations is introduced into the theory of unbounded derivations in operator algebras. A useful result for the phase transition theory will be shown for these derivations.

Generalized phase-space descriptions as linear representations of operators

Abstract: The generalized phase-space descriptions of a quantum system are constructed as special linear representations of the space of the linear operators, acting on the state vector space of the system. The relationship between quantum mechanics and classical mechanics is studied in terms of the phase-space descriptions.

On reductive algebras containing compact operators

Abstract: It is shown that a reductive operator algebra containing an injective compact operator is selfadjoint. A subalgebra T of the algebra of bounded linear operators on a Hilbert space is reductive if it is weakly closed, contains the identity operator, and has the property that I is invariant under U whenever V}i is an invariant subspace of L (cf. [4]). The reductive algebra problem (raised in [3]) is the question: Is every reductive algebra a von Neumann algebra? An affirmative answer to this question would be a very powerful result which would imply, in particular, that every operator has an invariant subspace (cf. [3], [4]). Partial results on the reductive algebra problem have been found by a number of people: The known results are all discussed in [4]. The purpose of this note is to record the observation that Lomonosov's remarkable result [1] implies another special case of the reductive algebra problem. W. B. Arveson, Carl Pearcy, Heydar Radjavi, Allen Shields (and undoubtedly others) have observed that the following lemma is the essence of Lomonosov's result, and that the lemma shows, in particular, that a transitive algebra (i.e., an algebra whose only invariant subspaces are 10} and h) containing a nonzero compact operator is strongly dense. I am grateful to Carl Pearcy for describing these results to me. A full discussion is contained in [4]. Lomonosov's Lemma [1]. If U is a transitive algebra of operators (not necessarily closed in any topology), and if K is any nonzero compact operator, then there exists an A 6 21 such that 1 is in the point spectrum of AK. The result on reductive algebras which can easily be obtained using this beautiful lemma is the following. Theorem. A reductive algebra which contains an injective compact operator is a von Neumann algebra. Received by the editors May 14, 1973. AMS (MOS) subject classifications (1970). Primary 46L15, 47A15, 47B05. lResearch supported by National Research Council of Canada Grant A5211. Copytight

On the geometry of noncommutative spectral theory

Abstract: We shall consider an order unit space (A, e) and a base-norm space (V, K) in separating order and norm duality with A pointwise monotone a-complete, i.e. for every descending sequence {an} inA + there exists a G A such that (a, x) = limn(an,x) for x GK. (See [1] for definitions and proofs.) We write M G TA if M is a weakly closed supporting hyperplane of A and F = C\\ {ME TA\\F C M} for F C A . (One may think of F as a \"minimal tangent space\" for A at F.) M is a smooth order ideal of A if M = (A + n Af)~, and F is a semiexposed face of A if F — A + O F. For a projection P: A —* 4̂ we write im P = 4̂ + Pi imP, ker P = A H ker P. Two projections P, Q: A —* 4̂ are quasi-complementary (q.c) if im (2 = ker P, ker Q = im P. Similar definitions apply with V in place of A. A weakly continuous positive projection P of A (or V) with ||P|| < 1 is smooth if ker P is a smooth order ideal. A projection R of V is neutral if \\\\Rv\\\\ = ||u|| implies Rv = u for u G F + . This term relates to physical filters which are \"neutral\" in that when a beam passes through with intensity undiminished Q\\Rv\\\\ = ||u||), then the filter is neutral to it (Rv = u).

Descending problem in Green's function approach to quantum field theory

Abstract: The question how to determine lower many-point functions in terms of higher ones, which we call the descending problem, is discussed for the (o4)1+3 model of quantum field theory. Equations to be considered are non-linear non-compact operator equations in complex Banach spaces.

Preliminary results in the Banach space formulation of master equation and subdynamics theory in quantum statistics

Abstract: The Liouville−von Neumann equation and a useful decomposition of the resolvent of the generator of the time evolution operators are obtained in the formulation of quantum statistics by means of the pair of Banach spaces (τc, B), where τc is the space of the ’’trace−class’’ operators and B is the space of all bounded operators, defined on a Hilbert space.

Approximate representations of a local current algebra

Abstract: An approximate method for dealing with nonrelativistic many‐body quantum systems having short range interactions is developed using local currents. The scheme is based on determining approximate representations of subalgebras of the local currents. This mathematical framework is used to discuss several approximation schemes.

Some problems in functional analysis

Abstract: The first part of this thesis is a study of strongly continuous (or weakly) continuous one parameter semigroups and groups of bounded linear maps on Banach spaces, their perturbation and scattering, paying particular attention to operator algebras. CHAPTER 1. We develop the spectral theory of derivations and groups of automorphisms on operator algebras, and study their relationships. The spectrum of a bounded strongly continuous one parameter group on a Banach space is characterised in terms of the spectrum of its infinitesimal generator. The relationship between Arveson's and Borchers' approach to the spectral theory of groups of *‑automorphisms on operator algebras is brought out, and we consider the problem of unitary implementation. CHAPTER 2. Following S.C. Lin's reflexive Banach space theory, we consider the perturbation, similarity and scattering of strongly continuous one parameter semigroups and groups on Banach spaces, using a time dependant approach and a strong type of smoothness, with particular reference to C*‑algebras and preduals of W*‑algebras. Some results regarding time evolution in quasi local algebras are used to derive scattering of quasi free evolution groups in the CAR algebra by inner perturbations. CHAPTER 3. The previous chapter leads us to study scattering of ultraweakly continuous groups of σ-weakly continuous linear maps on von Neumann algebras, by smooth perturbations of a certain weak type. We give a technique for handling groups of *‑automorphisms. CHAPTER 4. We study time dependant perturbations and scattering of strongly continuous one parameter groups on a Banach space E. The problem is raised to the higher space Lp(ℝ;E) (1 ≤ p ≤ ∞) or C0(ℝ;E), where the perturbation is made time independant and the methods of Chapter 2 apply. We characterise certain strongly continuous groups on the higher space in terms of propogators on the lower space, and show how their scattering is related. Some examples are given. The second part of the thesis investigates the structure of completely positive maps. CHAPTER 5. Schwartz type inequalities for n-positive linear mappings on *‑algebras are obtained. We demonstrate why the bounded completely positive linear maps for a Banach *‑algebra with approximate identity, and in particular for a C*‑algebra, should be regarded as higher order state spaces, A theorem of F. and M. Riesz is generalised to give a sufficient condition for the covariance of certain representations of a C*‑algebra relative to a one parameter group of *‑automorphisms. A completely positive analogue of Tomiyama's theorem regarding the singularity of conditional expectations on W*‑algebras is obtained. We characterise the completely positive linear mappings on the CCR algebra. Operator algebra analogues of Nagy's hilbert space dilations and Stroescu's Banach space dilations are obtained. The infinitesimal generators of strongly continuous one parameter semigroups and groups of linear maps with various positivity properties are studied. CHAPTER 6. Unbounded completely positive linear maps or operator valued weights on C*‑algebras are defined. We construct the Stinespring representation for an unbounded completely positive linear map α. We study the natural order structure for such maps, and following van Daele for scalar valued weights, when α has dense domain, we construct a largest operator valued weight α0 majorised by α, and with the property that it is the upper envelope of continuous completely positive maps. Following Combes we study the quasi equivalence, equivalence and type of the Stinespring representation associated with operator valued weights. Following van Daele, we study an unbounded completely positive map α which is invariant under a group of *‑automorphisms and has dense domain, and construct a G-invariant projection map φ of the set ℑ of continuous completely positive maps dominated by α, onto the set ℑ0 of G invariant elements of ℑ. This is used to derive various properties of the envelope of ℑ0. Asymptotically abelian systems and their ergodic bounded completely positive maps are studied. Observations are made regarding the possible classification of the spectral type of strongly continuous one parameter groups of *‑automorphisms of a C*‑algebra which possess an invariant operator valued weight.

C-Number Representation for Multilevel Systems and the Quantum-Classical Correspondence

Abstract: We use the r-mode direct product coherent states as generating functions for r-level coherent states which are ideal for the description of collective behavior of an ensemble of N identical r-level atoms or molecules, where the r-levels are not necessarily evenly spaced. It is noted that the Lie algebra for an r-level system can be given a realization in terms of bilinear combinations of boson creation and annihilation operators. This provides a homomorphism from the algebra describing a multimode system to the algebra describing a multilevel system. This in turn provides a homomorphism from the multimode coherent states and their diagonal projectors onto the multilevel coherent states and their diagonal projectors. The action of a creation or annihilation operator on a multimode projector can be replaced by the action of a first order differential operator.