Showing papers on "Algebra representation published in 1972"

Modern applied algebra

Abstract: Introduction Definition of a Graph, Simple Graph, Konigsberg bridge problem, Utilities problem, Finite and Infinite graphs, Regular graph, Matrix representation of graphs Adjacency matrix, Incidence matrix and examples; Paths and Circuits Isomorphism, Sub graphs, Walk, Path, Circuit, Connected graph, Euler line and Euler graph; Operations on graphs Union of two graphs, Intersecton of two graphs and ring sum of two graphs; Hamiltonian circuit, Hamiltonian path, Complete graph, Traveling salesmen problem. Trees and fundamental circuits, cutsets.

A correspondence between Hopf ideals and sub-Hopf algebras

Abstract: In this paper we show a bijective correspondence between normal Hopf ideals and sub-Hopf algebras of a commutative Hopf algebra over a field k. This gives a purely algebraic proof of the fundamental theorem [2, III, §3, no7] of the theory of affine k-groups.

On central division algebras

Abstract: For any integern such that 8|n or for which there exists an odd primeq such thatq 2|n, there is a central division algebra of dimensionn 2 over its center which is not a crossed product. The algebra constructed in this paper is the algebraQ(X 1,…,X)m, the algebra generated over the rationalQ bym(≧2) generic matrices.

A note on quadratic Jordan algebras of degree 3

Abstract: McCrimmon has defined a class of quadratic Jordan algebras of degree 3 obtained from a cubic form, a quadratic mapping and a base point. The structure of such an algebra containing no absolute zero-divisor is determined directly. A simple proof of Springer's Theorem on isomorphism of reduced simple exceptional quadratic Jordan algebras is given.

The Field algebra and its positive linear functionals

Abstract: We show that positive linear functionals on the field algebra are necessarily continuous and can be represented by conical measures. Furthermore extension theorems for continuous linear functionals, defined on a subspace of the field algebra, to positive linear functionals are discussed.

Cohomologies of parabolic lie algebras

Abstract: Ifp is a parabolic subalgebra of a semisimple algebra, the cohomology ofp is trivial in the associated representation.

Gel'fand-Kirillov dimension for algebras associated with the Weyl algebra

Abstract: Let K be a commutative field of characteristic zero. The Weyl algebra of degree n over K is defined as the associative algebra An (K) whose generators satisfy the canonical commutation relations for a system with n degrees of freedom. It is shown that if B lies in a certain class of subalgebras of the quotient field of An (K), then where A is a subalgebra of B, C (A, B) the commutant of A in B and DimK denotes a dimension introduced by Gel’fand and Kirillov [1]. Applications and possible generalizations of this result are discussed. RÉsUMÉ. Soit K un corps commutatif de caracteristique 0. L’algèbre de Weyl d’indice n sur K est définie comme l’algèbre associative An (K) à generateurs satisfaisant les relations de commutation a n degrés de liberte. Au cas où B est continue dans une certaine classe de subalgèbres du corps des fractions de An (K), il est démontré que où A est une subalgèbre de B, C (A, B) le commutant de A dans B, et Dimp est une dimension définie par Gel’fand et Kirillov [1]. On examine les utilisations et les généralisations eventuelles de ce résultat. (*) Work supported by the National Science Foundation under N. S. F. Grant No. GP-31359. (**) Permanent address : Department of Physics and Astronomy, Tel-Aviv University, Ramat Aviv, Israel.

Primitive ideals of *-algebras associated with transformation groups

Abstract: We extend results of E. G. Effros and F. Hahn concerning their conjecture that if (G, Z) is a second countable locally compact transformation group, with G amenable, then every primitive ideal of the associated C*-algebra arises as the kernel of an irreducible representation induced from an isotropy subgroup. The conjecture is verified if all isotropy subgroups lie in the center of G and either (a) the restriction of each unitary representation of G to some open subgroup contains a one-dimensional subrepresentation, or (b) G has an open abelian subgroup and orbit closures in Z are compact and minimal.

Equivalence of projections

Abstract: THEOREM: An A W*-algebra is the ring generated by its projections if and only if it has no abelian summand. COROLLARY: Every equivalence in an A W*-algebra may be implemented by a partial isometry in the ringgenerated by the projections of the algebra. The corollary is extended to certain finite Baer *-rings. An early proposition in the theory of von Neumann algebras is that every such algebra is the closed linear span of its projections. Dixmier observed that in a factorial algebra, the passage to the closure can be dispensed with if ring multiplication is also allowed: Every factorial von Neumann algebra is the algebra (sums, products and scalar multiples are the allowable operations) generated by its projections [2, Proposition 7]. Fillmore and Topping removed the restriction on the center as much as it can be removed: A von Neumann algebra is the algebra generated by its projections if and only if its abelian summand (if it has any) is finite-dimensional [8]; inspection of their arguments shows that the result is valid for A W*-algebras. What about the ring (differences and products the allowable operations) generated by the projections ? The answer is favorable, but abelian summands must be banished altogether: THEOREM 1. An A W*-algebra is the ring generated by its projections if and only if it has no abelian summand. COROLLARY. Every equivalence in an A W*-algebra may be implemented by a partial isometry in the ring generated by the projections of the algebra. The following remarks review the definitions and motivate the results. It is instructive to enlarge the setting somewhat; in the following remarks, A is a Baer *-ring [11], that is, an involutive ring such that the right annihilator of each subset is the principal right ideal generated by a projection (=selfadjoint idempotent). 1. Projections e,f in A are said to be equivalent (relative to A), written e'-'f, if there exists wcA such that w*w=e and ww*=f; such an element Received by the editors July 8, 1971. AMS 1969 subject classiflcations. Primary 4665.

Embedding rational division algebras

Abstract: Necessary and sufficient conditions are given for two K-division rings, K an algebraic number field, to have precisely the same set of subfields. Using this, an example is presented of two K-division rings having precisely the same set of subfields such that only one of the division rings can be embedded in a Q-division ring. Let K be a field. By a K-division ring we mean a finite-dimensional division algebra with center K. If D is a K-division ring and k is a field, k'c K, we say that D is k-adequate if D can be embedded in a k-division ring. Similarly, if L is a field, we say that L is k-adequate if L is a subfield of some k-division ring. Clearly, if D is k-adequate then so is every subfield of D. In [4] the converse was raised: if every subfield of D is k-adequate, must D be k-adequate? We show that the answer to this question is no by exhibiting two K-division rings D1 and D2 having precisely the same set of subfields and such that D1 is k-adequate (and so every subfield of D2 is also k-adequate) but D2 is not k-adequate. Throughout this paper K will denote an algebraic number field. We will use freely the classification theory of K-division algebras by means of Hasse invariants. The reader is referred to [3] for the relevant theory. If a is a prime of K and D is a K-division ring, we denote the Hasse invariant of D at Y by inv,, D. The order of inv. D in Q/Z is denoted by l.i. D. Here Q denotes the field of rational numbers and Z is the ring of ordinary integers. We denote the completion of K at the prime 9 by K,. The dimension of D over K is denoted by [D: K]; we use the same notation for the dimension of field extensions. We begin by establishing criteria for two K-division rings to have precisely the same set of subfields. THEOREM 1. Let D1 and D2 be K-division rings. Then D1 and D2 have precisely the same set of subfields if and only if l.i.g Dj=l.i.,+ D2 for all primes 9 of K. Received by the editors July 19, 1971. AMS 1970 subject classifications. Primary 16A40; Secondary 12A65.

On reflexive operator algebras

Abstract: Let be a weakly closed algebra of operators in a Hilbert space , containing a maximal commutative -subalgebra of the algebra of all bounded linear operators in . One investigates the problem of the reflexivity of (an operator algebra is said to be reflexive if it contains every operator for which all invariant subspaces of the algebra are invariant). It is proved that each of the following two conditions is sufficient for the reflexivity of : a) the lattice of the invariant subspaces of is symmetric; b) the algebra is generated by minimal projectors.One obtains other results too, referring to more general problems.Bibliography: 8 titles.

Continuity and linearity of centralizers on a complemented algebra.

Abstract: Let A be a semisimple complemented algebra and let T be a mapping of A into itself such that either T(xy) = xTy or T(xy) = (Tx)y holds for all x, y E A. If T is defined everywhere on A then T is a bounded linear operator. 1. A right centralizer on an algebra A is a mapping T of A into A such that T(xy) = xTy for all x, y E A. A left centralizer is a mapping T: A -> A such that T(xy) = (Tx)y for all x, y E A. This terminology is somewhat different from the terminology of [5] and is due to B. Johnson, who developed the theory of centralizers in [3] and was able to show that for a certain class of Banach algebras centralizers are always linear and bounded. In [4] he showed that this is the case when the algebra has a certain type of bounded approximate identity. The purpose of this paper is to extend these results of Johnson to the case of complemented algebras. Results of [4] are not applicable to our case. In fact the authors are convinced that Johnson's condition on existence of a certain type of approximate identity in the case of complemented algebras would be equivalent to the assumption of finite-dimensionality of the algebra. Inasmuch as every proper right H*-algebra [8] is a complemented algebra, we have extension of Johnson's result to all types of H*-algebras, and in particular, to the algebra of Hilbert Schmidt operators. We developed our theory for right centralizers but it is obvious that the same theory could be developed for left centralizers. 2. In this section we recall some basic definitions and facts from the theory of complemented algebras. For a more complete background the reader is referred to [6] and [7]. Presented to the Society, October 31, 1970; received by the editors August 19, 1970 and, in revised form, January 10, 1971. AMS 1970 subject classifications. Primary 46K1 5; Secondary 47B10.

The algebra generated by physical filters

Abstract: This paper investigates mathematical properties of a finite-dimensional real algebra of linear operators which are generated by an orthomodular lattice of filters in the sense of Mielnik [4]. Properties of filter decomposability and a representation theorem for the vector space underlying the algebra mentioned are derived.