Showing papers on "Idempotence published in 1969"

On the Wedderburn principal theorem

Abstract: Introduction. Let A be a strictly power-associative algebra with radical iV and such that the difference algebra A -N is separable. Then we say that A has a Wedderburn decomposition if A has a subalgebra SA N with A = S+ N (vector space direct sum). It is known that associative [1], alternative [15] and Jordan [3], [13] algebras have Wedderburn decompositions. In addition, if A is commutative and A-N separable with no nodal subalgebras such that either the simple summands of A -N have degree > 3 or A is stable, then A has a Wedderburn decomposition [5]. For our purposes, an algebra is a finite dimensional vector space on which a multiplication is defined which satisfies both distributive laws. We define xl = x and xk+l= xkx. An algebra A is called power-associative if xax1 = xa + B for all positive integers ac and 3 and every x in A. An algebra is called strictly power-associative if AK is power-associative for every scalar extension K of the base field. As a consequence of [11], if char #&2, 3, 5 then a power-associative algebra is strictly power-associative. In this paper, the radical N of A is the maximal nil ideal and a nonnil algebra with zero radical is said to be semisimple. An algebra is separable if it is semisimple over every scalar extension of the base field. We will call an algebra simple if it is semisimple and contains no proper ideals. The associator (x, y; z) = (xy)z x(yz) and the commutator (x, y) = xy -yx. An algebra is nodal if each element can be written as ac 1 + z with z nilpotent where the set of nilpotent elements is not a subalgebra. The center of A is the set of all elements that commute and associate with all of A. It is known [16, p. 16] that the center of a simple algebra is a field and is a finite extension of the base field. For char = 2, define x y = (xy +yx)/2 and define A + as the vector space A with multiplication defined by x y. When char 7 2, it has been proved [2] that if A is power-associative and if e is an idempotent (e2 = e & 0) then (1) A=Ae(1)+Ae(l12)+Ae(O) where Ae(t)={X: x.e=tx}. In addition, we have [2]: (2) Ae(l)Ae(O) =Ae(O)Ae(l) = 0, (3) xe=ex=O for x in Ae(O),

Measure algebras and functions of bounded variation on idempotent semigroups

Abstract: Our main result establishes an isomorphism between all functions on an idempotent semigroup S with identity, under the usual addition and multiplication, and all finitely additive measures on a certain Boolean algebra of subsets of S, under the usual addition and a convolution type multiplication. Notions of a function of bounded variation on S and its variation norm are defined in such a way that the above isomorphism, restricted to the functions of bounded variation, is an isometry onto the set of all bounded measures. Our notion of a function of bounded variation is equivalent to the classical notion in case S is the unit interval and the "product" of two numbers in S is their maximum.

Rigidity of generalized uniserial and Frobenius algebras

Abstract: Introduction. In [2], [4] Gerstenhaber, Nijenhuis and Richardson introduced the concept of deformations of an algebra A over a field. In this note we consider the question of whether there are any reasonable classes of algebras which admit nontrivial deformations but for which all the deformations remain in the class. The main theorem states that if A is a generalized uniserial basic algebra, then every deformation of A is generalized uniserial. The theorem actually gives a complete description of all deformations of a generalized uniserial basic algebra. The second theorem states that the class of Frobenius algebras is a class closed under deformations. We begin by setting up the notation. Throughout A will denote an associative algebra over a field k which admits a Wedderburn decomposition, A =S+N where S is K separable and N is the Jacobson radical. A will be called a basic algebra if the simple components of A are one dimensional over k and in this case we will write S = 1 kei where ei is the identity of the ith component of S. A generalized uniserial algebra A is an algebra such that for any primitive idempotent e the left (resp. right) modules Ae (eA) have a unique decomposition series. Following [2 ], A o =A 0 kk((t)) where k((t)) is the field of quotients of the power series ring over k in one indeterminant t. By a deformation of A we will mean an associative multiplication induced by a bilinear function ft: A 0 kA--Ao of the form

Idempotent Noether lattices

Abstract: In his paper, Abstract commutative ideal theory [2 ], Dilworth proved that a Noether lattice on which the multiplication is the meet operation is a finite Boolean algebra. This note proves that if the multiplication in a Noether lattice is idempotent (A2= A for all A in the lattice), then the lattice is a finite Boolean algebra. In a Noether lattice the term maximal element refers to a maximal nonidentity element (i.e. a coatom).

Commutative Banach Algebras with Idempotent Maximal Ideals

Abstract: Let be a commutative Banach algebra over the complex field C , M an ideal of . Denote by M 2 the set of all finite linear combinations of products of elements from M . M will be termed idempotent if M 2 = M . The purpose of this paper is to investigate the structure of commutative Banach algebras in which all maximal ideals are idempotent.

Commuting projections with assigned ranges

Abstract: 1. Let I be a Hilbert space. A projection is a bounded idempotent linear operator. Orthogonal projections form a proper subclass of the class of all projections. A problem is to get a condition for there to exist commuting projections E1, * E,n with the same ranges as given orthogonal projections P1, * , P,n respectively. This will be settled in terms of properties of the sublattice, generated by P1, * * , P,, in the lattice of all orthogonal projections. In case n = 2, commuting projections with minimum norms are constructed. Another problem is to find commuting projections E1, * * , E,n in a Hilbert space S, containing I as a subspace, such that Pjx =PEjx for xCeSD, j= 1, 2, ... , n, where P is the orthogonal projection from 9 onto t. This will be proved to be always possible.