M. Cabrera

EXTENDED CENTROID AND CENTRAL CLOSURE OF MULTIPLICATIVELY SEMIPRIME ALGEBRAS

Let A be an algebra and let M(A) denote the (unital) multiplication algebra of A. In (1) it was proved that if both A and M(A) are prime algebras, then the extended centroids of A and M(A) are isomorphic, and moreover Q(M(A)) is isomorphic to M(Q(A)), where Q(.) means central closure. In this paper we generalize these results to the case in which A is a semiprime algebra and we avoid any initial condition on M(A) by using the theory of Martindale algebras of symmetric quotients relative to filters of denominators given by K. McCrimmon in (2). In fact we prove these results in a more general setting by including operator algebras on A that contain the multiplication algebra of A.

Lie Quotients for Skew Lie Algebras

Let A be a semiprime associative algebra with an involution ∗ over a field of characteristic not 2, let KA be the Lie algebra of all skew elements of A, and let ZKA denote the annihilator of KA. The aim of this paper is to prove that if Q is a ∗-subalgebra of Qs(A) (the Martindale symmetric algebra of quotients of A) containing A, then KQ/ZKQ is a Lie algebra of quotients of KA/ZKA. Similarly, [KQ, KQ]/Z[KQ,KQ] is a Lie algebra of quotients of [KA,KA]/Z[KA,KA].

A note on real H*-algebras

Complex (associative) H*-algebras were introduced and studied in detail by Ambrose[ 1 ]; it was proved that every complex H*-algebra with zero annihilator is the l 2 -sum of a suitable family of topologically simple complex H*-algebras and that the H*-algebras ( H ) of all Hilbert-Schmidt operators on any complex Hilbert space H are the only topologically simple complex H*-algebras. In a recent paper [ 2 ] Balachandran and Swaminathan observe that the reduction of the theory of real H*-algebras to the topologically simple case follows easily with minor changes of the complex argument, and they prove a theorem describing topologically simple real H*-algebras. This theorem can be equivalently reformulated as follows.

Multiplicative Semiprimeness of Skew Lie Algebras

Abstract Let A be a semiprime associative algebra with an involution over a field of characteristic not 2, let K be the Lie algebra of all skew elements of A, and let Z [K, K] denote the annihilator of the Lie algebra [K, K]. We will prove that the multiplication algebra of the semiprime Lie algebra [K, K]/Z [K, K] is also semiprime. As a consequence, the multiplication algebra of [K, K]/Z [K, K] is prime, whenever [K, K]/Z [K, K] is prime. We will obtain similar results for the Lie algebra K/Z K whenever the base field has characteristic zero.

Multiplicative-Semiprimeness of Nondegenerate Jordan Algebras

Abstract We show that the multiplication algebra of a nondegenerate Jordan algebra is a semiprime algebra.

FUNCTIONAL IDENTITIES IN JORDAN ALGEBRAS: ASSOCIATING MAPS

ABSTRACT We prove that if J is a prime nondegenerate Jordan algebra and if F is an additive map from J into the unital central closure of J that satisfies for each , where denotes the associator, then there exist and an additive map such that for each , where denotes the extended centroid of J.

π-Complemented Algebras Through Pseudocomplemented Lattices

For an ideal I of a nonassociative algebra A, the π-closure of I is defined by

A Characterization of π-Complemented Algebras

\overline{I} = {\rm Ann}({\rm Ann} (I))

π-Complementation in the Unitisation and Multiplication Algebras of a Semiprime Algebra

, where Ann(I) denotes the annihilator of I, i.e., the largest ideal J of A such that IJ = JI = 0. An algebra A is said to be π-complemented if for every π-closed ideal U of A there exists a π-closed ideal V of A such that A = U ⊕ V. For instance, the centrally closed semiprime ring, and the AW∗-algebras (or more generally, boundedly centrally closed C∗-algebras) are π-complemented algebras. In this paper we develop a structure theory for π-complemented algebras by using and revisiting some results of the structure theory for pseudocomplemented lattices.

Closed Prime Ideals in Algebras with Semiprime Multiplication Algebra

π-complemented algebras are defined as those algebras (not necessarily associative or unital) such that each annihilator ideal is complemented by other annihilator ideal. Let A be a semiprime algebra. We prove that A is π-complemented if, and only if, every idempotent in the extended centroid of A lies in the centroid of A. We also show the existence of a smallest π-complemented subalgebra of the central closure of A containing A. In the case that A is a C*-algebra, this subalgebra turns out to be a norm dense *-subalgebra of the bounded central closure of A. It follows that a C*-algebra is boundedly centrally closed if, and only if, it is π-complemented.