Showing papers on "Operator algebra published in 1979"

Duality theory for covariant systems

Abstract: If (A, p, G) is a covariant system over a locally compact group G, i.e. p is a homomorphism from G into the group of *-automorphisms of an operator algebra A, there is a new operator algebra W called the covariance algebra associated with (A, p, G). If A is a von Neumann algebra and p is a-weakly continuous, W is defined such that it is a von Neumann algebra. If A is a C*-algebra and p is norm-continuous W will be a C*-algebra. The following problems are studied in these two different settings: 1. If W is a covariance algebra, how do we recover A and p? 2. When is an operator algebra W the covariance algebra for some covariant system over a given locally compact group G? Introduction. If G is a locally compact group and p: G -Aut(A) is a continuous homomorphism of G into the group of *-automorphisms of an operator algebra A, the triple (A, p, G) is called a covariant system. (A more precise definition is given in Chapters 2 and 3.) This notation was introduced by Doplicher, Kastler and Robinson in 1966, [10], but already Murray and von Neumann considered special cases with A abelian and G discrete in constructing the first non-type I factors. Covariant systems have turned out to be very interesting objects both in theoretical physics and in mathematics. With a representation of (A, p, G) we shall mean a pair (S, U) consisting of a unitary representation U of G and a *-representation S of A with S and U operating over the same Hilbert space such that Spx(a)' UxSaUxI fora&A,xe G. Doplicher, Kastler and Robinson showed that the representation theory of (A, p, G) was essentially the same as that of a certain operator algebra 9l called the covariance algebra of (A, p, G). The representation theory of W has been extensively studied by M. Takesaki in [22], G. Zeller-Meier in [30] and E. G. Effros and F. Hahn in [11] among others, and the covariance algebras provide us with a rich variety of examples of operator algebras. For instance many examples of factors are obtained this way (cf. [20, Chapter 4.2] for some), and A. Connes and M. Takesaki have recently shown that all type III Received by the editors March 13, 1975. AMS (MOS) subject classifications (1970). Primary Primary 46L05, 46L10.

Nonselfadjoint crossed products (invariant subspaces and maximality)

Abstract: Let 2 be the von Neumann algebra crossed product determined by a finite von Neumann algebra M and a trace preserving automorphism. In this paper we investigate the invariant subspace structure of the subalgebra 2+ of 2 consisting of those operators whose spectrum with respect to the dual automorphism group on 2 is nonnegative, and we determine conditions under which 2+ is maximal among the a-weakly closed subalgebras of 2. Our main result asserts that the following statements are equivalent: (1) M is a factor; (2) 2+ is a maximal a-weakly closed subalgebra of 2; and (3) a version of the Beurling, Lax, Halmos theorem is valid for 2+. In addition, we prove that if W is a subdiagonal algebra in a von Neumann algebra e and if a form of the Beurling, Lax, Halmos theorem holds for X, then e is isomorphic to a crossed product of the form 2 and W is isomorphic to 2+. Introduction. Crossed products were introduced into operator theory by Murray and von Neumann in their first paper [14]. The algebras which they constructed as crossed products are now most commonly called group measure algebras. Subsequently, their construction was abstracted, generalized, and analyzed by numerous authors and it is fair to say that at present crossed products are ubiquitous in the theory of operator algebras. Indeed, Feldman and Moore [8] have recently shown that it is very likely that every von Neumann algebra can be realized as a crossed product-perhaps of a complicated nature. In this paper we consider von Neumann algebras which are constructed as very simple crossed products and focus our attention on certain nonselfadjoint subalgebras contained in them. Roughly speaking, the subalgebras we study stand in the same relation to the crossed products as the Hardy space H '(T), the space of boundary values of bounded analytic functions on the unit disc, stands in relation to the Lebesgue space L?(T). Received by the editors November 11, 1977. AMS (MOS) subject classifications (1970). Primary 46L15, 46L10, 46K05; Secondary 47C15.

Cylindrical lens systems described by operator algebra.

Abstract: In a recent work, an operator algebra was developed for the description of axially symmetrical coherent optical systems. The present work extends the operator representation to incorporate cylindircal lenses and is used to analyze the transforming properties of an arbitrarily oriented cylindrical lens. The results provide the basis for synthesizing various systems for signal processing, such as convolution and correlation operations.

C[*]- and W[*]-algebras symmetry groups decomposition of states

Abstract: (Volume 1).- C*-Algebras and von Neumann Algebras.- Groups, Semigroups, and Generators.- Decomposition Theory.- References.- Books and Monographs.- Articles.- List of Symbols.

Principal functions, index theory, geometric measure theory and function algebras

Abstract: The aim of this paper is twofold: first to provide evidence of a nice relationship between index theory and operator algebras within the framework of geometric measure theory by exhibiting basic examples involving one dimensional singular integral operators; second to expose certain connections that exist involving the principal function associated to an operator having trace class self-commutator and the theory of function algebras.

The carrier space of a reflexive operator algebra

Abstract: Many properties of nest algebras are actually valid for reflexive operator algebras with a commutative subspace lattice. In this paper we collect a number of such results related to the carrier space of the algebra. Included among these results are a generalization of Ringrose's criterion, a description of the partial correspondence between lattice homomorphisms of the carrier space and projections in the lattice, the construction of isometric representations of certain quotient algebras, and a direct sum decomposition of the commutant of the core modulo the intersection of the spectral ideals. Let J^ = Alg ^ where if is a commutative subspace lattice and let ^ be the intersection of all the spectral ideals in Jzf. (See §1 for definitions.) In §1 we generalize Ringrose's criterion to the commutative subspace lattice case: A e ^ if, and only if, for each e > 0 there is a finite family {Et} of mutually orthogonal intervals from & such that Σ^ = l and WE.AE.W < e, i = 1, . , n. We also prove that ^ is the closed linear span of commutators of the form AL — LA, where A e jy and L e £f. In § 2 we describe the partial correspondence between certain projections in Sf and certain lattice homomorphisms in the carrier space X. A necessary (but not sufficient) condition for an operator A to be in the radical of *$/ is given in §3. In §4 we exhibit isometric representations as algebras of operators acting on Hubert space of each quotient algebra J^fjj^φ and of the quotient *S%f\

Operators commuting with positive operators

Abstract: Necessary and sufficient conditions are obtained for an operator to commute with a positive operator. Throughout the paper, by an operator we mean a bounded linear transformation acting in a Hilbert space H. The algebra of all operators in H is denoted by B(H). Arveson's theorem [1] about transitive algebras states that if W is a strongly closed transitive algebra of operators and if W contains a maximal abelian selfadjoint algebra (with respect to B(H)), then W = B(H). (A transitive algebra is one whose only invariant subspaces are {0} and H.) Foia? [2] gives a different proof of Arveson's theorem mainly based on the following facts: (F 1) If W is a strongly closed proper subalgebra of B(H), then W leaves the range of a nonzero, noninvertible positive operator K invariant. In particular, if 9f contains a maximal abelian selfadjoint algebra ER, then K can be chosen such that K E NR and WK c KW. (F2) If W is a uniformly closed algebra and WK c KW for some noninvertible positive operator K # 0, then W is not transitive. In the proof of (F2) it is shown that if E is the resolution of the identity for K and T E 9X, then TE([t, oo))H c E([t/a, oo))H for 0 1, and if liminf(IIK-'TKnII/an) O then TE([t, oo))H c E([t/a, oo))H, 0 < t < 11K]I. (2) Note that we assume K-nTKn can be extended boundedly to all of H. In (1) and (2), K is an injective positive operator and E is its resolution of the identity. Conversely, Theorems 2 and 3 show that if T satisfies (2), then (1) holds but for a replaced by a2. As corollaries, we obtain necessary and Received by the editors May 15, 1978 and, in revised form, July 12, 1978. AMS (MOS) subject classifications (1970). Primary 47B15, 47C05.

On the matrix representation of closed linear operators: An extension of the Von Neumann’s theory

Abstract: The general theory of the matrix representation of operators in scalar product space is examined. It is proved that an extension of the Von Neumann’s theory on the matrix representation of closed symmetric operators in Hilbert space is possible for a larger class of closed operators. A necessary and sufficient condition for the existence of a matrix representation of operators in ’’Von Neumann’s sense’’ is given.

Fundamentals of Banach Algebras and C*-Algebras

Abstract: In this this first chapter, we lay the foundation for later discussion, giving elementary results in Banach algebras and C*-algebras. The first three sections are devoted to the general Banach algebras. The most important results in these sections are Theorem 2.5, Corollary 2.6, and Theorem 3.11, which are really fundamental in the theory of Banach algebras. Discussion of C*-algebras starts from Section 4. As an object of the theory of operator algebras, a C*-algebra is a uniformly closed self-adjoint algebra A of bounded linear operators on a Hilbert space ℌ. The major task of the theory of operator algebras is to find descriptions of the structure of {A,ℌ}. This problem splits into two problems: (a) Find descriptions of the algebraic structure of A alone; (b) Given an algebra A, find all possible pairs {B,ℜ} such that B is isomorphic to A as an abstract algebra.

The field representation of the spin-1/2 quantum field theory

Abstract: In the mathematical framework of a Grassmann algebra the Schrodinger picture of the quantized Dirac theory is constructed in a representation in which the field operator is diagonal. An application to a simple theory of interacting fields is given.

The Algebraic Structure of the Thirring Model

Abstract: We summarize the representation theory of the group SU(1,1) as needed for the massless Thirring model. Representations of the current operator algebra are given taking account of conformal covariance. The conformal covariance transformation behaviour of the Thirring field is investigated. The Haag-Araki-Kastler observable algebra of the Thirring field is reconstructed from the Wightman theory of this model.

The splitting of operator algebras. ii

Abstract: We say the singly generated C*-algebra, C*(Tt®Tt)9 splits if C*(TΊ φ Γ 2)=C*(Γ1) φ C*(T2). A necessary and sufficient condition is derived for the splitting of C*(Tλ φ T2) in terms of the topological structure of the primitive ideal space of C*CZ\ φ T2). In particular, when C*(2\ φ T2) is strongly amenable, the necessary and sufficient condition can be simplified and does not depend on the topology of the primitive ideal space of C*(TX φ Tz). Several applications of this theorem, such as the cases, among others, where Tl9 T2 are compact operators, and C*{Ti), C*(T2) have only finite-dimensional irreducible representations, are discussed. For the splitting of the T7*-aIgebra, W*{TX φ T2), two equivalent conditions are derived which are quite different in nature. It is also shown that WHTλφ T2) splits if either WHReTiφReT*) or W*(Im2\ φ ImT 2) splits, but the converse is false. An example is given to show that TF*(ΓiΦ Γ2) splits whereas C*(2Piφ T2) does not. l Introduction* Let Jzf be a C*-algebra. If S?/ has an