Showing papers on "Operator algebra published in 1983"
Book•
01 Jan 1983
TL;DR: In this article, the authors compare normal states and unitary equivalence of von Neumann algebras, including the trace and the trace trace of the trace of a projection.
Abstract: Comparison theory of projections--exercises and solutions Normal states and unitary equivalence of von Neumann algebras--exercises and solutions The trace--exercises and solutions Algebra and commutant--exercises and solutions Special representations of $C^*$-algebras--exercises and solutions Tensor products--exercises and solutions Approximation by matrix algebras--exercises and solutions Crossed products--exercises and solutions Direct integrals and decomposiitons--exercises and solutions Bibliography Index.
2,937 citations
01 Feb 1983
TL;DR: In this paper, it was shown that the rank-one subalgebra of a reflexive algebra with completely distributive invariant subspace lattice is strongly dense in strong, weak, ultrastrong or ultraweak topologies.
Abstract: It is shown that if the invariant subspace lattice of a reflexive algebra CT, acting on a separable Hilbert space, is both commutative and completely distributive, then the algebra generated by the rank-one operators of &T is dense in &T is any of the strong, weak, ultrastrong or ultraweak topologies. Some related density results are also obtained. The main purpose of this note is to clarify the role of the rank-one operators in a reflexive algebra with a (commutative) completely distributive invariant subspace lattice. A complete lattice is completely distributive if it permits distribution of the lattice operations over families ofarbitrary cardinality (see below for a precise definition). Every nest, i.e. every totally ordered subspace lattice, is completely distributive. For a subspace lattice, the property of being completely distributive is intimately related to the presence of rank-one operators in the associated reflexive algebra. For a given reflexive algebra C, we will refer to the subalgebra generated by the rank-one operators in d as the rank-one subalgebra of e. In [10] it was shown that if the rank-one subalgebra of d( is strongly dense in 6 then the lattice of invariant subspaces of ( is completely distributive. In the converse direction, it was known (see [9]) that the rank-one subalgebra of a reflexive algebra with completely distributive invariant subspace lattice was big enough to determine the lattice and the natural question to ask was: is it big enough to determine the algebra? Specifically, is the rank-one subalgebra of a reflexive algebra with completely distributive invariant subspace lattice strongly dense in the algebra? This was known to be true for nest algebras [2] and also in the case where the underlying Hilbert space was finite-dimensional. Subsequently, Lambrou [7] showed that complete distributivity of a subspace lattice implied a condition somewhat weaker than the desired strong density. Here we show that the answer is affirmative if the additional requirement of commutativity is imposed on the invariant subspace lattice. Some related density results are obtained. Specifically, our main result is that if the invariant subspace lattice of a reflexive algebra C6, acting on a separable Hilbert space, is commutative and completely distributive, then the rank-one subalgebra of C is dense in e in any of the strong, weak, ultrastrong or ultraweak topologies. As a consequence, we show that if the invariant subspace lattice of a reflexive algebra e is Received by the editors January 19, 1983. This paper was presented (by the first author) in the Special Session of Operator Algebras in Operator Theory, Annual Meeting of the A.M.S., January 1983. 1980 Mathematics Subject Classification. Primary 47D25. ?l 983 American Mathematical Society 0002-9939/83 $1.00 + $.25 per page
67 citations
TL;DR: In this article, the projective space of a complex Hilbert space H is considered both as a Kahler manifold and as the set of pure states of the von Neumann algebra B(H).
Abstract: The projective space of a complex Hilbert space H is considered both as a Kahler manifold and as the set of pure states of the von Neumann algebra B(H). A link is given between these two structures. Special attention is devoted to topology, orientation and automorphisms of the structures and Wigner's theorem.
66 citations
TL;DR: In this paper, the authors point out explicitly the canonical Hamiltonian structure of these equations, by introducing a suitable symplectic structure on the underlying phase space in various representations, leading to the Schr\"odinger equation, together with its complex conjugate, which can be recognized as a particular form of the Hamilton canonical equations in this frame.
Abstract: In previous work the approach to stochastic quantization, originally proposed by Nelson, has been formulated in the frame of the stochastic variational principles of control theory. Then the Hamilton-Jacobi-Madelung equation is interpreted as the programming equation of the controlled problem, to be associated with the hydrodynamical continuity equation. Here we point out explicitly the canonical Hamiltonian structure of these equations, by introducing a suitable symplectic structure on the underlying phase space in various representations. One possible representation leads to the Schr\"odinger equation, which, together with its complex conjugate, can be recognized as a particular form of the Hamilton canonical equations in this frame. Then a suitably selected time-invariant subalgebra of the classical hydrodynamical algebra, closed under Poisson bracket pairing, is shown to be connected to the standard quantum observable operator algebra. In this correspondence Poisson brackets for hydrodynamical observables become averages of quantum observables in the given state. From this point of view stochastic quantization can be interpreted as giving an explanation for the standard quantization procedure of replacing the classical particle (or field) observables with operators, according to the scheme $p\ensuremath{\rightarrow}(\frac{h}{i})\frac{\ensuremath{\partial}}{\ensuremath{\partial}x}$, $l\ensuremath{\rightarrow}(\frac{h}{i})\frac{\ensuremath{\partial}}{\ensuremath{\partial}\ensuremath{\varphi}}$, etc. This discussion shows also the relevance of the canonical symplectic structure of the quantum state space, a feature which seems to have been overlooked in the axiomatic approaches to quantum mechanics.
29 citations
TL;DR: In this article, a dilation theory for a class of contraction operators acting on a separable, infinite dimensional, complex Hilbert space was developed, in which the algebra of bounded linear operators on ~ is denoted by Y ( ~ ).
Abstract: 1. This note is a continuation of our earlier paper [-3], in which we developed a dilation theory for a certain class of contraction operators acting on a separable, infinite dimensional, complex Hilbert space ~ . The notation and terminology in what follows is taken from [3]. For the convenience of the reader we recall a few pertinent definitions. The algebra of bounded linear operators on ~ is denoted by Y ( ~ ) . If T~Se(2C), the ultraweakly closed algebra generated by T and l~e is denoted by dr; we recall that d r can be identified with the dual space of the quotient space Q r = ( z c ) / • where (zc) denotes the ideal of trace-class operators in 5~(24 ~) and • is the preannihilator of d r in (z c), under the pairing
29 citations
01 Jan 1983
TL;DR: In this article, it was shown that the weakly closed operator algebra generated by an equicontinuous a-complete Boolean algebra of projections on a quasi-complete locally convex space consists entirely of scalar-type operators.
Abstract: It is shown that the weakly closed operator algebra generated by an equicontinuous a-complete Boolean algebra of projections on a quasi-complete locally convex space consists entirely of scalar-type operators. This extends W. Bade's well-known theorem that the same assertion is valid for Banach spaces; however, the technique of proof here differs from his method, which extends only to metrizable spaces. In [1], W. Bade showed that every operator in the weakly closed operator algebra generated by a a-complete Boolean algebra of projections on a Banach space is a scalar-type spectral operator (in the sense of Dunford, [2]). The aim of this note is to show that this result can be extended to locally convex spaces. The methods used by Bade are not available, because they make use of the fact that the weak operator closure of the algebra generated by a a-complete Boolean algebra of projections on a Banach space is again an algebra; the corresponding assertion is false for locally convex spaces in general. These methods are replaced by the theory of integration with respect to closed spectral measures (see ?2). Let X be a locally convex Hausdorff space. The space X will always be assumed to be quasi-complete. Let L(X) denote the space of all continuous linear operators on X, equipped with the topology of pointwise convergence. THEOREM. Let the space L(X) be quasi-complete. Let ( be an equicontinuous, a-complete Boolean algebra of projections in L( X). Then every operator in the weak operator closure of the algebra generated by C is a scalar-type spectral operator. 1. Preliminaries. The dual space of X is denoted by X'. If q is a continuous seminorm on X, let Uq= {x' E X'; I (x, x') I for allx such that q(x) 1). An X-valued vector measure is a a-additive map m: G1 -X whose domain 9Z is a a-algebra of subsets of a set Q2. For each x E X', the complex-valued measure E + (m(E), x'), E E 6Z, is denoted by (m, x'). If q is a continuous seminorm on X, then the q-semivariation, q(m), of m is the set function defined by q(m)(E) = sup{I (m, x')| (E); x' E Uq}, E E 9AX. Received by the editors January 6, 1982 and, in revised form, April 16, 1982. 1980 Mathemautics Subject Classificatioz. Primary 47B40; Secondary 46G10, 47D30.
20 citations
20 citations
TL;DR: For quantum fields with trigonometric interaction in arbitrary space dimension, the authors constructed a representation of the Lorentz group by automorphisms on a Banach space generated by the Weyl algebra.
Abstract: For quantum fields with trigonometric interaction in arbitrary space dimension we construct a representation of the Lorentz group by automorphisms on a Banach space generated by the Weyl algebra.
19 citations
TL;DR: The theory of completely bounded maps between C*-algebras has been studied in this article, where Wittstock, Smith, Paulsen, and Huruya have reviewed and discussed their results.
Abstract: Recent development of the theory of completely bounded maps between C*-algebras such as those results by Wittstock, Smith, Paulsen and Huruya are reviewed and discussed as well as the author's results. In recent development of the theory of operator algebras it has been recognized that particularity of infinite dimensional noncommutative order structure is in their matricial order and appropriate positivity of linear maps compatible with this order structure should be complete positivity. Thus as counterparts of completely positive maps we have to be naturally concerned with completely bounded maps when we consider bounded linear maps on operator algebras closely related to their structure. Recent results such as Christensen [5] and Haagerup [6] give evidence of this fact in the sense that solutions of the problems are equivalent to complete boundedness of involved maps. In this paper we intend to review recent development of the basic theory of completely bounded maps on C*-algebras. We shall be however mainly concerned with the results for the class of all completely bounded maps. In Section 2 we describe the solution on a pair of C*-algebras between which every bounded map becomes completely bounded. Results in this section also mean that in most cases the subspace of completely bounded maps is quite thin in the whole space of bounded linear maps. We shall deal with the growth of the norms of multiplicity maps in Section 3, determining the case where the norms of nmultiplicity map rn always coincides with the completely bounded norm \\t\\cb. In this section Theorem 3.2 and Proposition 3.4 seem to be new. Section 4 is Received January 5, 1983. * Faculty of Science, Niigata University, Niigata 950-21, Japan.
TL;DR: In this article, it was shown that the GNS construction can be generalized to real B*-algebras containing an algebra *-isomorphic to the quaternion algebra by the use of quaternions linear functionals and Hilbert Q•modules.
Abstract: It is shown that the Gel’fand–Naimark–Segal (GNS) construction can be generalized to real B*‐algebras containing an algebra *‐isomorphic to the quaternion algebra by the use of quaternion linear functionals and Hilbert Q‐modules. An extension of the Hahn–Banach theorem to such functionals is proved.
TL;DR: The relation between a quantum dynamical semigroup and the associated quantum stochastic processes can be formulated by an analogy in operator algebras to the martingale problem of Stroock and Varadhan (1969) for classical diffusions as mentioned in this paper.
Abstract: The relation between a quantum dynamical semigroup and the associated quantum stochastic processes can be formulated by an analogy in operator algebras to the martingale problem of Stroock and Varadhan (1969) for classical diffusions. It is shown that under simple circumstances the formulation yields the explicit solution. The authors give three such examples.
TL;DR: The notion of absolute continuity for linear forms on B *-algebras is introduced and investigated in this paper, where the main result is a general Radon-Nikodym type theorem in the context of B*-algesbras which extends a well-known theorem of S. Sakai and has diverse applications in the theory of operator algebra.
Abstract: The notion of absolute continuity for linear forms onB
*-algebras is introduced and investigated. The main result is a general Radon-Nikodym type theorem in the context of B*-algebras which extends a well-known theorem of S. Sakai and has diverse applications in the theory of operator algebras.
01 Jan 1983
TL;DR: In this article, it was shown that there exists a nontrivial closed subspace of P&r which is both a lattice-ideal and an algebraideal of Sr, namely the space Cr = {A C pr: A I is compact}.
Abstract: Let jgr be the Banach algebra (and Banach lattice) of all regular operators on 12, i.e. the algebra of all operators A on 12 which are given by a matrix (amn) such that (I a,, 1) defines a bounded operator I A I. We show that there exists exactly one nontrivial closed subspace of P&r which is both a lattice-ideal and an algebra-ideal of Sr, namely the space Cr = {A C pr: A I is compact}. We also show that every nontrivial ideal in pr iS included in KCr It is well known that the only nontrivial closed (algebra) ideal in the Banach algebra X(H) of all bounded operators on a separable Hilbert space H is the ideal Y(H) consisting of all compact operators on H and that SC(H) includes every nontrivial ideal. In this note we prove order-theoretic analogues of these results. To give a precise statement we need some notation. Let H = 12, the Hilbert space of all (real or complex) square-summable sequences, and denote by {en}, n = 1, 2,..., the standard basis in 12. Every bounded operator A on 12 can be represented by a
TL;DR: In this article, the statistical sum of a degenerate elliptic functional is introduced for the construction of invariants of the type of the Ray-Singer torsion in quantum field theory.
Abstract: Applications of the theory of elliptic operators in quantum field theory are indicated. The concept of the statistical sum of a degenerate elliptic functional is introduced; this concept finds application both in quantum field theory and outside it (for the construction of invariants of the type of the Ray-Singer torsion).
01 Dec 1983
TL;DR: In this paper, the authors define a notion of local stabilizability for systems whose coefficients lie in a C*-algebra, and show that such a system is stabilizable if and only if it is locally stabilizable.
Abstract: We define below a notion of local stabilizability for systems whose coefficients lie in a C*-algebra, and we show that such a system is stabilizable if and only if it is locally stabilizable. The C*-algebra is not assumed to be commutative. Since B(H) is a C*- algebra, the results of this paper apply in particular to any system whose coefficients can be represented as bounded linear operators on some Hilbert space H.
TL;DR: Soit A une C*-algebre, soit p un polynome sur C et soit a dans M(A) tel que p(ada)=0.
Abstract: Soit A une C*-algebre, soit p un polynome sur C et soit a dans M(A) tel que p(ada)=0. On etudie l'existence d'un λ dans Z(M(A)) tel que p(α-λ)=0
TL;DR: In this paper, a physically evident requirement on asymptotic product states is formulated in a mathematically precise way and shown to fix the S-matrix uniquely for relativistic field theories of short-range interactions.
Abstract: A physically evident requirement on asymptotic product states is formulated in a mathematically precise way and shown to fix the S-matrix uniquely for relativistic field theories of short-range interactions.
01 Jan 1983
TL;DR: Several statistical or dynamical systems modelling a variety of physical phenomena, such as spin chains, two dimensional spin lattices and chemical crystals, or particles in δ function interaction, share the same momentous underlying structures, which are essentially the applicability of Bethe's superposition ansatz for wave functions, the commutativity of transfer matrices, and the existence of an important ternary operator algebra as discussed by the authors.
Abstract: Several statistical or dynamical systems modelling a variety of physical phenomena, such as spin chains, two dimensional spin lattices and chemical crystals, or particles in δ function interaction, share the same momentous underlying structures, which are essentially the applicability of Bethe’s superposition ansatz for wave functions, the commutativity of transfer matrices, and the existence of an important ternary operator algebra. Their close relationship among each other, and with features like integrability and S matrix factorization discussed in field theoretical context, became familiar in the course of time.
TL;DR: In this article, it was shown that in the Hilbert space of a quantum field theory with a nonzero mass gap there exists a dense set of vectors, each entire analytic for the energy-momentum operators, that are cyclic for the polynomial algebra P(Rd) for any nonempty O(⊆ Rd).
Abstract: It is shown that in the Hilbert space of a quantum field theory with a nonzero mass gap there exists a dense set of vectors, each entire analytic for the energy–momentum operators, that are cyclic for the polynomial algebra P(Rd) [and for the local polynomial algebras P(O), for any nonempty O ⊆ Rd]. It is proven that for every vector Φ from this dense set there exists an element Q ∈ P(Rd) such that QΦ=Ω, where Ω is the vacuum, and QΩ=0. A similar, stronger result is proven for free field theories (including mass zero).