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Showing papers on "Operator algebra published in 1980"


Journal ArticleDOI
TL;DR: In this paper, the authors present a class of C*-algebras and point out its close relationship to topological Markov chains, whose theory is part of symbolic dynamics.
Abstract: In this paper we present a class of C*-algebras and point out its close relationship to topological Markov chains, whose theory is part of symbolic dynamics. The C*-algebra construction starts from a matrix A =(A (i,j))i,~ z, Z a finite set, A(i,j)c{0, l}, and where every row and every column of A is non-zero. (That A(i,j)e{O, 1} is assumed for convenience only. All constructions and results extend to matrices with entries in 2~+. We comment on this in Remark 2.18.) A C*-algebra 6~ A is then generated by partial isometries Si~O(i~X ) that act on a Hilbert space in such a way that their support projections Qi=S*S~ and their range projections P~ =SIS* satisfy the relations

1,042 citations


Journal ArticleDOI
TL;DR: In this paper, the method of factorization of operators, which has been used to derive the Miura transformation of the KdV equation, is extended to some third-order scattering operators, and transformations between several fifth-order nonlinear evolution equations are derived.
Abstract: The method of factorization of operators, which has been used to derive the Miura transformation of the KdV equation, is here extended to some third‐order scattering operators, and transformations between several fifth‐order nonlinear evolution equations are derived. Further applications are discussed.

123 citations


Journal ArticleDOI
TL;DR: A characterization of state spaces of Jordan algebras by as mentioned in this paper is improved to a form with more physical appeal in the simplified case of a finite dimension, which is the case in this paper.
Abstract: A characterization of state spaces of Jordan algebras by Alfsen and Shultz is improved to a form with more physical appeal (proposed by Wittstock) in the simplified case of a finite dimension.

108 citations



Journal ArticleDOI
TL;DR: In this paper, an infinite class of finite-dimensional irreducible representations of Lie superalgebras of an arbitrary rank is constructed, and an orthonormal basis in the corresponding representation space is found, and matrix elements of the generators are calculated.
Abstract: An infinite class of finite‐dimensional irreducible representations and one particular infinite‐dimensional representation of the special linear superalgebra of an arbitrary rank is constructed. For every representation an orthonormal basis in the corresponding representation space is found, and the matrix elements of the generators are calculated. The method we use is similar to the one applied in quantum theory to compute the Fock space representations of Bose and Fermi operators. For this purpose we first introduce a concept of creation and annihilation operators of a simple Lie superalgebra and give a definition of Fock‐space representations.

41 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that normalized positive maps in algebras of the form A ⊗ A with A an abelian C∗-algebra can be described by a generalized Bochner theorem.

32 citations


Journal ArticleDOI
TL;DR: In this article, it is shown that quantum field theory can be formulated in such a way that the field and space variables are not distinguished a priori. Applications to supergravity are given.

26 citations


Journal ArticleDOI
TL;DR: In this article, a q-analogue of MacMahon's Master Theorem is given, and an elimination procedure for linear partial difference operators is demonstrated. But this procedure is restricted to combinatorics.
Abstract: The algebra of linear partial difference operators is investigated, and an elimination procedure demonstrated. Applications to combinatorics are given. In particular, a new proof and a q-analogue of MacMahon’s Master Theorem are given.

22 citations


Journal ArticleDOI
TL;DR: In this paper, it has been shown how the Heisenberg algebra can be generalized to produce an algebraic structure in which it is possible to describe space translations in a way analogous to the description of rotations in a Clifford algebra.
Abstract: It has been proposed that the implicate order can be given mathematical expression in terms of an algebra and that this algebra is similar to that used in quantum theory. In this paper we bring out in a simple way those aspects of the algebraic formulation of quantum theory that are most relevant to the implicate order. By using the properties of the standard ket introduced by Dirac we describe in detail how the Heisenberg algebra can be generalized to produce an algebraic structure in which it is possible to describe space translations in a way that is analogous to the description of rotations in a Clifford algebra. This approach opens up the possibility of going beyond the limits of the present quantum formalism and we discuss briefly some of the new implications.

18 citations


Journal ArticleDOI
TL;DR: Davidson's construction of a Hilbert space and of quantum operators on the basis of the Fenyes-Nelson stochastic mechanics is extended to the case in which a dissipative force linear in the velocity is present as mentioned in this paper.

17 citations


Journal ArticleDOI
TL;DR: In this article, the authors study derivations on unbounded operator algebras in connection with those in operator algesbras and show that a derivation with some range-property on a left EW#-algebra induced by an unbounded Hilbert algebra is strongly implemented by an operator which belongs to an algebra of measurable operators.
Abstract: This paper is a study of derivations on unbounded operator algebras in connection with those in operator algebras. In particular we study spatiality of derivations in several situations. We give the characterization of derivations on general *-algebras by using positive linear functionals. We also show that a derivation with some range-property on a left EW#-algebra induced by an unbounded Hilbert algebra is strongly implemented by an operator which belongs to an algebra of measurable operators.

Journal ArticleDOI
TL;DR: In this article, the authors studied spectral subspaces under the action of compact groups on Banach spaces, with particular reference to operator algebras, and showed that the spectral subspace can be characterized by compact groups.

Journal ArticleDOI
TL;DR: In this article, it was shown that the elements of the closed operator algebra generated by one-dimensional singular integral operators with piecewise continuous coefficients with a fixed finite set of points of discontinuity can be written as the sum of a singular integral operator, a compact operator, and generalized Mellin convolutions.
Abstract: It is shown that the elements of the closed operator algebra generated by one-dimensional singular integral operators with piecewise continuous coefficients with a fixed finite set of points of discontinuity can be written as the sum of a singular integral operator, a compact operator, and generalized Mellin convolutions. Their Gohberg-Krupnik symbol is given in terms of the Mellin transform. This gives an explicit construction of an operator with prescribed Gohberg—Krupnik symbol.

Journal ArticleDOI
TL;DR: A *-polarization is a linear submanifold of the space of C∞ functions on phase space as mentioned in this paper, which connects phase space mechanics to the usual operator formulation of quantum theory.
Abstract: The method of *-polarization connects phase space mechanics to the usual operator formulation of quantum theory. A *-polarization is a linear submanifold of the space of C ∞ functions on phase space. Elements of a *-polarization are in direct correspondence with the Schroedinger wave functions and this correspondence induces the Weyl correspondence between classical observables and operators. All generalized Moyal algebras admit *-polarizations and a general method is thus available for translating *-quantization into operator language.

Journal ArticleDOI
Bent Ørsted1
TL;DR: In this paper, a nonlinear analog to the free quantum field of Bose statistics is presented, in which the linear one-particle space is replaced by a non-linear infinite-dimensional Hermitian symmetric space D, and the quantum field is constructed as a Hilbert space of holomorphic functions on D.



Journal ArticleDOI
TL;DR: In this paper, a state space theory for time-varying state trajectories is presented, which is based on the approach of Schnure and Steinberger, and it is shown that the Hilbert space is the direct integral integral integral of the state spaces.

Journal ArticleDOI
TL;DR: In this article, an iterative method which determines the low-lying eigenvalues of a Hermitian operator defined in a finite-dimensional vector space is extended to a specific type of unbounded Hermitians in a Hilbert space.
Abstract: An iterative method which determines the low-lying eigenvalues of a Hermitian operator defined in a finite-dimensional vector space is extended to a specific type of unbounded Hermitian operator defined in a Hilbert space. As an illustrative numerical example the extended algorithm has been applied to the quantum mechanical harmonic oscillator problem.

Journal ArticleDOI
01 Aug 1980
TL;DR: In this paper, it was shown that the C*-crossed product G Xa 9C can be identified with a certain C *-subalgebra of the W*crossed products G x a D.
Abstract: To each W*-dynamical system (T, G, a) corresponds canonically a C*-dynamical system (9VRc, G, aI'D1t). We show that the C*-crossed product G Xa 9C can be identified with a certain C*-subalgebra of the W*-crossed product G x a DThe major part of the theory of noncommutative dynamical systems and their crossed products is Takesaki's work; see e.g. [7] and [8]. An important contribution, however, was made by Landstad who in [1] characterized those operator algebras that are crossed products with a given locally compact group G. Landstad's theory of G-products for abelian groups was exploited in [4] and [2] and we shall use it again to solve a problem arising from the difference between C*and W*-crossed products. A general exposition of noncommutative dynamical systems can be found in Chapters 7 and 8 of [5], but only the elementary parts of the theory will be needed here. Recall that a triple (6(, G, a) is called a C*-dynamical system if &6 is a C*-algebra and a is a representation of the locally compact abelian group G as automorphisms on &, such that each function t >a,(x), x E &, is norm continuous. If 6IT is a von Neumann algebra we define analogously a W*-dynamical system (6OR, G, a), but now only with the requirement that each function t -a(x), x E 'DX, is a-weakly continuous. Given a W*-dynamical system ('1., G, a) define DUZC to be the set of elements x in 6R for which the function t -* at(x) is norm continuous, see [5, 7.5.1]. Clearly DU is a G-invariant C*-subalgebra of 6Th containing all elements of the form a/f(y) = fat(y)f(t) dt, y E 6Th,f E L1(G) (since translation is continuous on L1(G)). Using an approximate unit in L1(G) we see that DIU is in fact generated by elements af(y), and therefore a-weakly dense in 69R. Thus we obtain from (6Th, G, a) a canonically defined C*-dynamical system (91Xc, G, a1lTc). We shall study the relation between the W*-crossed product G x a 6Th and the C*-crossed product G x a 9ZD. Recall from [8, ?4] (cf. [5, 7.10.3]) that to each W*-dynamical system (9T, G, a) we can construct the dual system (G xa 9, G, a). We may identify 9Th with the von Neumann subalgebra of G xa 9Th consisting of the fixed points for G under Received by the editors April 5, 1979 and, in revised form, August 17, 1979. AMS (MOS) subject classifications (1970). Primary 46L05; Secondary 46L10.

Journal ArticleDOI
TL;DR: In this paper, the authors studied multiplication operators in L 2 -spaces of matrix measures as models for self-adjoint operators of finite multiplicity and derived a version of the state space isomorphism theorem.
Abstract: The paper studies multiplication operators in $L^2$-spaces of matrix measures as models for self -adjoint operators of finite multiplicity. The module theoretic aspects are emphasized and an analysis of intertwining maps, that is, module homomorphisms relative to the algebra of multiplication operators by bounded Borel functions, is given. Finally the machinery is applied to the study of dynamical systems withself-adjoint generators. Controllability aspects are studied and a version of the state space isomorphism theorem is derived.


01 Oct 1980
TL;DR: In this paper, the irrational rotation algebra A θ is considered and there are non approximately bounded pregenerators δ i (i = 1, 2) of A ǫ with the same domain such that given a ∗ -derivation δ of A ë with D ( δ ) = D (ǫ i ), there exist (k, l )∈ R 2 and an approximately bounded ∗ −derivation ϴ 3 of Aǫ such that δ = kδ 1 + lδ 2 + δ 3,
Abstract: Abstract Let A θ be the irrational rotation algebra. Then there exist non approximately bounded pregenerators δ i ( i = 1, 2) of A θ with the same domain such that given a ∗ -derivation δ of A θ with D ( δ ) = D ( δ i ), there exist ( k , l )∈ R 2 and an approximately bounded ∗ -derivation δ 3 of A θ such that δ = kδ 1 + lδ 2 + δ 3 , which can be considered as a solution of a problem of Sakai for two dimensional space quantizations.

Book ChapterDOI
TL;DR: In this paper, the authors describe the different aspects of the operator algebras and generalized orderings and show that a preordered vector space is a real vector space endowed with a reflexive and transitive relation that is compatible with the linear structure.
Abstract: Publisher Summary This chapter describes the different aspects of the operator algebras and generalized orderings. In the description of structure in real Banach spaces, certain abstract order relations are used to give an intrinsic characterization of the corresponding operator algebras. The idea of describing any given commutative operator algebra as the center with respect to a sufficiently general order structure on the underlying vector space might, besides its unifying aspect, turn out to be fruitful also for the investigation of properties of the operator algebra itself. A preordered vector space is a real vector space endowed with a reflexive and transitive relation that is compatible with the linear structure. A preordered algebra has always to be almost commutative and approximately a vector lattice. It is found that cone is algebraically and order isomorphic to a dense subalgebra of the ordered algebra of all continuous real valued functions on some compact Hausdorff space.