Showing papers on "Algebra representation published in 1979"

The second dual of a Banach algebra

Abstract: Let A be a Banach algebra over a field IF that is either the real field IR or the complex field ℂ, and let A' be its first dual space and A" its second dual space. R. Arens in 1950, gave a way of defining two Banach algebra products on A" , such that each of these products is an extension of the original product of A when A is naturally embedded in A" . These two products mayor may not coincide. Arens calls the multiplication in A regular provided these two products in A" coincide. Perhaps the first important result on the Arens second dual, due essentially to Shermann and Takeda, is that any C*-algebra is Arens regular and the second dual is again a C*-algebra. Indeed if A is identified with its universal representation then A" may be identified with the weak operator closure of A-hat. In a significant paper Civin and Yood, obtain a variety of results. They show in particular that for a locally compact Abelian group G ,Ll(G) is Arens regular if and only if G is finite. (Young showed that this last result holds for arbitrary locally compact groups.) Civin and Yood also identify certain quotient algebras of [Ll(G)]". Pak-Ken Wong proves that A-hat is an ideal in A" when A is a semi-simple annihilator algebra, and this topic has been taken up by S. Watanabe to show that [L 1 (G)]-hat is ideal in [L 1 (G) ]" if and only if G is compact and [M (G)]-hat is an ideal in [M(G)]" if and only if G is finite. One shoulu also note in this context the well known fact that if E is a reflexive Banach space with the approximation property and A is the algebra of compact operators on E, (in particular A is semi-simple annihilator algebra) then A" may be identified with BL(E). S.J. Pym [The convolution of functionals on spaces of bounded functions, Proc. London Math. Soc., (3) 15 (1965)] has proved that A is Arens regular if and only if every linear functional on A is weakly almost periodic. A general study of those Banach algebras which are Arens regular has been done by N.J. Young and Craw and Young. But in general, results and theorems about the representations of A" are rather few. In Chapter One we investigate some relationships between the Banach algebra A and its second dual space. We also show that if A" is a C*-algebra, then * is invariant on A. In Chapter Two we analyse the relations between certain weakly compact and compact linear operators on a Banach algebra A, associated with the two Arens products defined on A". We clarify and extend some known results and give various illustrative examples. Chapter Three is concerned with the second dual of annihilator algebras. We prove in particular that the second dual of a semi-simple annihilator algebra is an annihilator algebra if and only if A is reflexive. We also describe in detail the second dual of various classes of semi-simple annihilator algebras. In Chapter Four, we particularize some of the problems in Chapters Two and Three to the Banach algebra 𝓁1 (S) when S is a semigroup. We also investigate…

The algebraic and geometric classification of associative algebras of dimension five

Abstract: The scheme Alg5 of associative, unitary algebra structures on k5, k an algebraically closed field with char (k)≠2 is investigated. We establish the list of GL5-orbits on Alg5 under the action of structural transport. The numberalg5 of irreducible components of Alg5 is 10; a list of generic structures is included. We exhibit upper and lower bounds for the asymptotic behaviour of the numberalgn.

W-module structure in the primitive spectrum of the enveloping algebra of a semisimple Lie algebra

Abstract: Formulae developed to give a positive answer to Dixmier's problem for Verma and principal series submodules are used to show that each primitive ideal in the enveloping algebra of a semisimple Lie algebra identifies with a left ideal in the group algebra of the Weyl group. The possible behaviour of these left ideals under right multiplication leads to a conjecture for the set of order relations in the primitive spectrum.

Indecomposable division algebras

Abstract: (1979). Indecomposable division algebras. Communications in Algebra: Vol. 7, No. 8, pp. 791-817.

Jordan rings with nonzero socle

Abstract: Let Jbe a nondegenerate Jordan algebra over a commutative associative ring $ containing j. Defining the socle @ of \\ to be the sum of all minimal inner ideals of J, we prove that @ is the direct sum of simple ideals of j. Our main result is that if J is prime with nonzero socle, then either (i) J. is simple unital and satisfies DCC on principal inner ideals, (ii) $• is isomorphic to a Jordan subalgebra $-' of the plus algebra A+ of a primitive associative algebra A with nonzero socle S, and %' contains 5 +, or (iii) % is isomorphic to a Jordan subalgebra %\" of the Jordan algebra of all symmetric elements H of a primitive associative algebra A with nonzero socle S, and %\" contains H n S. Conversely, any algebra of type (i), (ii), or (iii) is a prime Jordan algebra with nonzero socle. We also prove that if Jis simple then % contains a completely primitive idempotent if and only if either \\ is unital and satisfies DCC on principal inner ideals or f is isomorphic to the Jordan algebra of symmetric elements of a »-simple associative algebra A with involution * containing a minimal one-sided ideal.

Linear maps of *-algebras preserving the absolute value

Abstract: In order that a linear map of 6'-algebras «¡>: & -» ® preserve absolute values, it is necessary and sufficient that it be 2-positive and preserve zero products of positive elements: if x and y are positive in &, with xy 0, then <&x)$(y) = 0. The generalized Schwarz inequalities of Kadison and Choi are extended to the nonunital case.

Properties of an Associative Algebra of Tensor Fields. Duality and Dirac Identities

Abstract: An algebra of forms in Minkowski space has been constructed. A multiplication between forms is defined as an extension of the quaternionic multiplications. The algebra obtained is associative with respect to this multiplication of order 16. Duality is expressed as (new) multiplication by a basis element. Vector identities in the algebra lead to a number of new trace identities. A new derivative operator expresses the four Maxwell equations in an especially transparent form.

Commutators in Banach algebras

Abstract: In a recent paper ( 6 ) the present author has shown that, for an element a of a Banach algebra A , the condition for all x ∈ A and some constant α is equivalent to [ x , a ]∈Rad a for all x ∈ A ; it turns out that α may be replaced by |α|σ It is the purpose of the present note to investigate a related condition

Hermitian Operators on Banach Jordan Algebras

Abstract: In this note, we examine some of the properties of Hermitian operators on complex unital Banach Jordan algebras, that is, those operators with real numerical range. Recall that a unital Banach Jordan algebra J, is a (real or complex) Jordan algebra with product a ˚ b, having a unit 1, and a norm ∥·∥, such that J, with norm ∥·∥, is a Banach space, ∥1∥ = 1, and, for all a and b in j,

The Algebra of Weyl Symbols and the Cauchy Problem for Regular Symbols

Abstract: An algebra of generalized functions and an isomorphic algebra of continuous linear operators in Schwartz space are defined. The properties of this algebra are studied, and the exponential function of certain elements of it is constructed. Bibliography: 16 titles.

Classification and construction of finite dimensional irreducible representations of the graded algebras; application to the (Sp(2n); 2n) algebra

Abstract: A method, which enables us to construct the finite‐dimensional representations of the graded Lie algebras [explicitly the GLA (Sp(2N); 2N)] on the irreducible tensors is suggested. Those tensors are constructed with the aid of the specifically symmetrized products of vectors from the fundamental [i.e., (2N+1) ‐dimensional] representation space of the graded Lie algebra. The tensors, on which it is possible to represent the graded algebra (Sp(2N); 2N) irreducibly, represent a generalization of tensors which are known from the general representation theory of the symplectic Lie algebra Sp(2N). The knowledge of the irreducible tensors of the algebra Sp(2N) gives us then the possibility of solving the problems of classification as well as construction of the irreducible tensors of the graded algebra (Sp(2N); 2N). For illustration, by using the suggested method of tensors the irreducible representations of the simplest graded algebra, i.e., of the algebra (Sp(2); 2) are constructed.

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\

On Banach algebra elements of thin numerical range

Abstract: Among the elements of a complex unital Banach algebra the real subspace of hermitian elements deserves special attention. This forms the natural generalization of the set of self-adjoint elements in a C*-algebra and exhibits many of the same properties. Two equivalent definitions may be given: if W ( h ) ⊂ , where W ( h ) denotes the numerical range of h (7), or if ║ e i λ h ║ = 1 for all λ ∈ . In this paper some related subsets are introduced and studied. For δ ≥ 0, an element is said to be a member of if the condition is satisfied. These may be termed the elements of thin numerical range if δ is small.