Showing papers on "Operator algebra published in 1981"
01 Jan 1981
TL;DR: A bale and grain feeder device includes an elongate conveyor and table structure which is mounted on the mixer-grinder vehicle chassis and cooperates with the conveyor paddles to convey the slabs of hay along the table where the hay or grain is directed to the hammer mill by paddles.
Abstract: The first edition of this book appeared in 1981 as a direct continuation of Lectures of von Neumann Algebras (by S.V. Stratila and L. Zsido) and, until 2003, was the only comprehensive monograph on the subject. Addressing the students of mathematics and physics and researchers interested in operator algebras, noncommutative geometry and free probability, this revised edition covers the fundamentals and latest developments in the field of operator algebras. It discusses the group-measure space construction, Krieger factors, infinite tensor products of factors of type I (ITPFI factors) and construction of the type III_1 hyperfinite factor. It also studies the techniques necessary for continuous and discrete decomposition, duality theory for noncommutative groups, discrete decomposition of Connes, and Ocneanu's result on the actions of amenable groups. It contains a detailed consideration of groups of automorphisms and their spectral theory, and the theory of crossed products.
415 citations
Book•
01 Jan 1981
TL;DR: In this paper, Wigner and Racah operators are associated with W-Algebra, an algebra of invariant operators, and a list of symbols Indices is given.
Abstract: 1. Introduction 2. Algebraic structures associated with Wigner and Racah operators 3. Null space properties and structure theorems for RW-Algebra 4. W-Algebra: an algebra of invariant operators 5. Special topics Appendix List of symbols Indices.
276 citations
78 citations
TL;DR: In this article, the ergodic and KMS channels are studied by operator algebraic methods and the dynamical properties of these channels are discussed. But the quantum ergodics are not considered.
71 citations
TL;DR: In this paper, the relationship between classical and quantum theories is expressed in the same mathematical language, in terms of a matrix algebra in a phase space, which makes clear the similarities of the two theories, but also certain essential differences, and lays a foundation for understanding their relationship.
Abstract: We approach the relationship between classical and quantum theories in a new way, which allows both to be expressed in the same mathematical language, in terms of a matrix algebra in a phase space. This makes clear not only the similarities of the two theories, but also certain essential differences, and lays a foundation for understanding their relationship. We use the Wigner-Moyal transformation as a change of representation in phase space, and we avoid the problem of “negative probabilities” by regarding the solutions of our equations as constants of the motion, rather than as statistical weight factors. We show a close relationship of our work to that of Prigogine and his group. We bring in a new nonnegative probability function, and we propose extensions of the theory to cover thermodynamic processes involving entropy changes, as well as the usual reversible processes.
57 citations
01 Mar 1981
TL;DR: In this article, it was shown that if si is cyclic, there is a *-representation 0: A -* B(H) and a bounded one-to-one positive operator P such that PO(a) = 7i(a),P.
Abstract: Let 7r: A -* B(H) be a bounded homomorphism of a C*-algebra into the bounded operators on a Hilbert space. We prove that, if si is cyclic, there is a *-representation 0: A -* B(H) and a bounded one-to-one positive operator P such that PO(a) = 7i(a)P. We include applications to 0-derivations and invariant operator ranges for operator algebras.
33 citations
TL;DR: In this paper, the Fourier operator description of Fourier optics is extended and applied to holography, and a compact expression for the description of the holographically reconstructed field distribution at an arbitrary plane.
Abstract: The operator description of Fourier optics is extended and applied to holography. The existing lens models for ideal holographic processes appear as a self-evident intermediate result; generalization to include apertures, recording-material modulation transfer function, and extended source effects is straightforward. The extended source effect is generally shown to be equivalent to a modification of the actual holographic apertures. The final result is a compact expression for the description of the holographically reconstructed field distribution at an arbitrary plane. A useful, comprehensive list of operator relations is given in two appendixes.
30 citations
01 Jan 1981
26 citations
Book•
01 Jun 1981
TL;DR: In this article, a C*-Algebra approach to the Cowen-Douglas Theory is presented, which is based on periodic distribution groups and isomorphisms of Automorphism Groups of Type II factors.
Abstract: On Closed Operator Algebras Generated by Analytic Functional Calculi.- A Conjecture Concerning the Pure States of B(H) and a Related Theorem.- A C*-Algebra Approach to the Cowen-Douglas Theory.- On Periodic Distribution Groups.- On the Smoothness of Elements of Ext.- Triviality Theorems for Hilbert Modules.- Exact Controllability and Spectrum Assignment.- Generalized Derivations.- Commutants Modulo the Compact Operators of Certain CSL Algebras.- Similarity of Operator Blocks and Canonical Forms. II. Infinite Dimensional Case and Wiener-Hopf Factorization.- Unitary Orbits of Power Partial Isometries and Approximation by Block-Diagonal Nilpotents.- Isomorphisms of Automorphism Groups of Type II Factors.- A Spectral Residuum for Each Closed Operator.- Two Applications of Hankel Operators.- A Rohlin Type Theorem for Groups Acting on von Neumann Algebras.- Derivations of C*-Algebras which Are Invariant Under an Automorphism Group.- Remarks on Ideals of the Calkin-Algebra for Certain Singular Extensions.- Modelling by L2-Bounded Analytic Functions.- The Maximal Function of Doubly Commuting Contractions.- Remarks on Hilbert-Schmidt Perturbations of Almost - Normal Operators.- Derivation Ranges: Open Problems.
14 citations
TL;DR: The speckle pattern produced by laser light scattered from a moving diffuse surface is described with linear system analysis and the mathematical procedures are greatly simplified by operator algebra.
Abstract: The speckle pattern produced by laser light scattered from a moving diffuse surface is described with linear system analysis. The mathematical procedures are greatly simplified by operator algebra. Results cited in the literature are shown to be special cases, derived under simplifying assumptions, from the general expressions obtained in the present work.
TL;DR: In this article, a generalized Weyl correspondence is defined between random variables on one side and linear operators in a separable Hilbert space on the other, and a relation among states on H (considered as positive nuclear operators on H) and the distribution functions of the random variables is shown.
Abstract: A so‐called generalized Weyl correspondence is defined among random variables on one side and linear operators in a separable Hilbert space H on the other. Besides such a correspondence, there is a relation among states on H (considered as positive nuclear operators on H) and the distribution functions of the random variables. By adding some new assumptions, several relations are shown. Later, we study two particularly interesting cases. In the first we connect dichotomic random variables with number operators in a Grassmann algebra H, and nuclear operators on H with probability measures in the set of all sequences made up of zero and one. In the second case we relate states between stochastic and quantum electrodynamics.
01 Jan 1981
TL;DR: In this article, the authors propose a method to solve the problem of "uniformity" and "uncertainty" in the context of health care, and propose a solution.
Abstract: §
TL;DR: In this paper, it was shown that a closed *-derivation in a C*-algebra is bounded if the range is contained in the domain of the C* algebra.
Abstract: The early study of derivations on operator algebras and Banach algebras concerned the continuity of them. Silov [21] showed that there is no norm on ~f~([0, 1]) under which ~ ( [ 0 , 1]) becomes a Banach algebra. In connection with this result, Kaplansky posed a question whether every derivation on a semi-simple Banach algebra is continuous. Johnson and Sinclair [11] showed that the answer is affirmative (see, [13, 19, 22]). In Sect. 2, we shall give a generalization of the Silov theorem, which is stated as follows: Let fi be a closed *-derivation in a C*-algebra 9.I. Suppose that ~ = ~ ~(~") is dense in 9.I. I f there exists a norm on ~ under which ~| becomes n>= l a Banach algebra, then 6 is bounded on 9.I. As a corollary, we shall show that a closed *-derivation in a C*-algebra is bounded if the range is contained in the domain.
TL;DR: In this article, the authors studied the Fredholm theorem of a C*-algebra of O-order pseudo-differential operators on L 2(ℝn).
Abstract: In this paper we study the Fredholm thoery of a C*-algebraOl of o-order pseudo-differential operators on L2(ℝn). IfK denotes the ideal of all compact operators of L2, the algebraOl will be generated by (i) the idealK, (ii) a function algebra CS(ℝn) and (iii) by the bounded operators xjΛ, DjΛ, j=1,...,n, Λ= H−1/2, H=1+¦x¦2−Δ. We show thatOl/K is a commutative C*-algebra with identity and obtain its Gelfany space M. This provides Fredholm criterion and index formula for a graded algebra of partial differential operators including all oeprators with polynomial coefficients. We also give Fredholm criterion and index formula for systems of such operators.
TL;DR: In this article, the trace class of operators on a Hilbert space is characterized in terms of existence of certain centralizers, i.e., the existence of a centralizer for each operator.
Abstract: The trace-class (τc) of operators on a Hilbert space is characterized in terms of existence of certain centralizers.
01 Jan 1981
TL;DR: In this article, the authors investigate closed operator algebras generated by analytic functional calculi for n-tuples of commuting operators which are decomposable or quasi-decomposable.
Abstract: In this note we investigate closed operator algebras generated by analytic functional calculi for n-tuples of commuting operators which are decomposable or quasi-decomposable. In particular, we obtain semisimplicity criteria which generalize a corresponding result for the closed full algebra generated by a spectral operator resp. by a decomposable operator due to U. Fix-man and L. Tzafriri resp. F.-H. Vasilescu.