Showing papers on "Operator algebra published in 1978"

Haar measure for measure groupoids

Abstract: It is proved that Mackey's measure groupoids possess an analogue of Haar measure for locally compact groups; and many properties of the group Haar measure generalize. Existence of Haar measure for groupoids permits solution of a question raised by Ramsay. Ergodic groupoids with finite Haar measure are characterized. Introduction. In their pioneering investigation of operator algebras on Hilbert space, Murray and von Neumann [12] obtained examples of non-type I factors using ergodic actions of a group on a measure space. Since then their method has been adapted and generalized several times to yield interesting new examples of von Neumann algebras; but despite recasting, the procedure has always seemed special and somewhat mystifying. The present work is the first of several papers treating Haar measure and convolution algebras of functions on George Mackey's measure groupoids [10]. This research permits us to interpret the construction made by Murray and von Neumann, as well as subsequent generalizations, in terms of these convolution algebras. Our unified treatment includes Krieger's construction of factors from nonfree actions of countable groups [7], Dixmier's examples of quasi-unitary algebras [2], and the regular representation of second countable locally compact groups. They all arise from modular Hilbert algebras of functions on some appropriate groupoid. Our approach seems natural because each step can easily be related to the special case of groups, which is widely known. Consider, for example, the case of a group action. Let g be a locally compact second countable group with Haar measure h; and suppose (S, ,) is a standard finite measure space on which g acts so that ,t remains invariant. We denote by sx the transform of s E S by x E g and assume that (s, x) + sx: S x g -> S is Borel measurable. The measure space (S X g, ju X h) becomes a measure groupoid with unit (or object) space S if we define the product (s, x)(t, y) = (s, xy) whenever t = sx. If f and g are suitably restricted functions on S x g, Dixmier [2] and Glimm [4] have used a product Received by the editors May 5, 1976 and, in revised form, April 13, 1977. AMS (MOS) subject classifications (1970). Primary 28A65, 28A70; Secondary 28-02. 'Preparation supported by NSF Grant MPS 74-19876 ? American Mathematical Society 1978 This content downloaded from 157.55.39.253 on Wed, 08 Jun 2016 05:24:39 UTC All use subject to http://about.jstor.org/terms

The Coupled Representation Matrix of the Pair Hamiltonian.

Abstract: A complete treatment of the spin hamiltonian for a pair of exchange coupled metal ions is given. The results obtained through the use of the Wigner-Eckart theorem are unrestricted with respect to the relative orientation of the single-ion tensors, and appropriate for both the S 1 = S 2 and S 1 ≠ S 2 cases. The necessary constants are derived so that matrix elements within a given spin multiplet can be treated by ordinary operator algebra with the coupled representation state vectors, |SM>, as a basis set. Explicit algebraic formulae for matrix elements between spin multiplets are presented for the first time. Symmetry restrictions are discussed in general and illustrated for the simple case of two ions related by an inversion centre, pair symmetry Ci .

The mathematical foundations of quantum theory

Abstract: Publisher Summary This chapter discusses the relation between mathematics and the quantum theory of physics After the development of noncommutative algebra, it was connected with dynamics by means of an analogy between the commutator and the Poisson bracket of the Hamiltonian form of mechanics and thus, a general quantum mechanics was set up An important feature of the theory was the wave equation, which had to be linear, and thus treated the time dimension differently from the space coordinates With the fundamental principle of quantum mechanics—that is, the principle of the superposition of states—the states of any quantum system provide a representation of the Poincare group, which is a mathematical representation of the quantum theories The study of the group requires working with the infinitesimal operators of which 10 are independent operators where 4 are translation operators and 6 are rotation operators They satisfy certain definite commutation relations Any representation of the Poincare group provides the independent operators satisfying these commutation relations Conversely, any set of 10 operators satisfying these relations gives a representation of the Poincare group

An asymptotic double commutant theorem for c*-algebras

Abstract: An asymptotic version of von Neumann's double commutant theorem is proved in which C*-algebras play the role of von Neumann algebras. This theorem is used to investigate asymptotic versions of simi- larity, reflexivity, and reductivity. It is shown that every nonseparable, norm closed, commutative, strongly reductive algebra is selfadjoint. Applications are made to the study of operators that are similar to normal (subnormal) operators. In particular, if T is similar to a normal (subnormal) operator and «■ is a representation of the C*-algebra generated by I, then ir(T) is similar to a normal (subnormal) operator. 1. Introduction. One of the reasons for the success of the theory of von Neumann algebras is J. von Neumann's double commutant theorem (46), which gives an alternate description of the weak closure of a selfadjoint algebra of operators. It is the purpose of this paper to prove an asymptotic version of the double commutant theorem that gives an alternate description of the norm closure of a selfadjoint algebra of operators. This asymptotic double commutant theorem helps to unify asymptotic versions of various operator-theoretic concepts (e.g., similarity, reflexivity, reductivity). Applications are made to the study of operators that are similar to normal (or subnormal) operators. Also a proof is given that a strongly reductive, nonseparable, commutative, norm closed algebra of operators is selfadjoint. Throughout, H denotes a separable, infinite-dimensional complex Hubert space, B(H) denotes the set of operators (bounded linear transformations) on H, and %(H) denotes the set of compact operators on H. Also § denotes a separable, nonempty subset of B(H). However, in §8 the separability assumptions on H and S will be dropped. If § Q B(H), then §* = {S*: S G S}, #"(§>) is the norm closed algebra generated by 1 and §, &W(S) is the weakly closed algebra generated by 1 and S, C*(§) is the C*-algebra generated by 1 and S, and W*(S ) is the von Neumann algebra generated by

Operator algebras and statistical mechanics

Abstract: Some topics in quantum statistical mechanics related to theory of operator algebras are reviewed.

Kaplansky-Hilbert Modules and the Self-Adjointness Of Operator Algebras

Abstract: It is shown that every weakly closed operator algebra having the same invariant linear manifolds as a von Neumann algebra must be self-adjoint. There is no restriction on the dimension of the underlying Hilbert space. The proof depends on a density theorem for algebras of bounded homomorphisms acting on a Kaplansky-Hilbert module. This use of Kaplansky-Hilbert modules provides a means of avoiding direct integral theory.

Quantum dynamics in the effective system states basis. I. Mathematical structure and time‐dependent perturbation theory

Abstract: A molecule in the presence of a radiation field is described by a non‐Hermitian reduced effective Hamiltonian and its eigenvectors, the effective system states. These states form a natural basis for the evaluation of spectroscopic properties; its use requires a generalized formulation of quantum dynamics. The mathematical structure of the vector space defined by those states is investigated; that vector space is found to be a Banach space which reduces to Hilbert space if the energies of the effective system states become real. Time reversal symmetry is found to be an important feature of the overall problem. The generalized time development operator and the Schrodinger equation which defines it are deduced, and the effect of that operator on the effective system states is discussed. The proper generalization of time‐dependent perturbation theory in the interaction picture is then derived and its use in applications is discussed in general terms.

A theory of classical limit for quantum theories which are defined by real Lie algebras

Abstract: A theory of classical limit is developed for quantum theories, the basic observables of which correspond to elements in some real Lie algebra L0. For both quantum and classical systems based on L0 the basic observables are contained in a unique universal algebra. This is the universal enveloping algebra U for the quantum case, and a universal commutative Poisson algebra L for the classical case. U and L are connected by a system of contraction maps. For certain sequences of representations and of vector states defined by them renormalized expectation values of the quantum variables are shown to converge to values of the corresponding classical variables at some point in the classical phase space. The classical phase space is obtained as a limit of certain systems of coherent states. The general theory is illustrated by several examples and counterexamples.

Invariant Operator Ranges

Abstract: This report describes a portion of joint work with M. Radjabalipour, H. Radjavi and P. Rosenthal [8]. The set of Hilbert space operators that leave invariant a fixed dense operator range is given a matrical representation. Also it is shown that every operator on an infinite dimensional Hilbert space has an uncountable collection of infinite dimensional invariant operator ranges such that any two of them have only the vector 0 in common.

C*-Algebras and Applications

Abstract: The theory of C*-algebras dates from the discovery by Gelfand and Naimark that uniformly closed self-adjoint operator algebras on Hilbert space—unlike the rings studied by von Neumann and Murray—could be characterized in simple intrinsic algebraic terms, independently of their action on Hilbert space. This opened up the study of the algebraic isomorphism classes of such algebras, in the sense of emphasizing its cogency. It was soon found that C*-algebras have certain applications in quantum mechanics, and especially in quantum field theory, in parts of group representation theory, and some other areas, in which W*-algebras could not be substituted.

Operator algebra and the stationary states of stellar magnetic fields

Abstract: A vector-operator algebra technique for solving magnetic field problems in a toroidal/poloidal representation is illustrated with physical examples. Among the illustrative examples are calculations of necessary and/or sufficient conditions for the existence of stationary magnetic fields in stellar interiors.

Pointwise ergodic theory on operator algebras

Abstract: We extend Birkhoff’s pointwise ergodic theorem from classical mechanics to the overlap with quantum mechanics.

On the representation of linear operators in L2 spaces by means of ’’generalized matrices’’

Abstract: This paper concerns the representation of linear operators of L2 spaces by means of ’’generalized matrices’’ as it is usual, following Dirac, in quantum mechanics and in electronics. The known possibility of representing (on a nuclear test‐function space) the bounded operators by means of distribution kernels is shown to extend to all the closable operators whose domain contains the test‐function space (hence to all the Hermitian operators whose domain contains the Schwartz space D of the infinitely differentiable functions with compact support). The representation of the adjoint operator is considered, and the possibility of representing the product of operators by means of a suitably defined ’’Volterra convolution’’ is studied. In particular it is shown that ‐algebras of unbounded operators (which are, for instance, generated by the canonical coordinates and momenta and the total energy of most quantum mechanical systems of n particles) may be represented isomorphically by means of distribution kernels,...