A. R. Villena

The structure of Lie derivations on C*-algebras

We prove that every Lie derivation on a C � -algebra is in standard form, that is, it can be

The Noncommutative Singer–Wermer Conjecture and ϕ-Derivations

The question of when a

Characterizing Jordan maps on C*-algebras through zero products

phi

Continuity of Lie Isomorphisms of Banach Algebras

n -derivation on a Banach algebra has quasinilpotent values, and how this question is related to the noncommutative Singer–Wermer conjecture, is discussed.

Symmetric amenability and Lie derivations

Let A and B be C *-algebras, let X be an essential Banach A -bimodule and let T : A → B and S : A → X be continuous linear maps with T surjective. Suppose that T(a)T(b) + T(b)T(a) = 0 and S(a)b + bS(a) + aS(b) + S(b)a = 0 whenever a, b e A are such that ab = ba = 0. We prove that then T = w Φ and S = D + Ψ, where w lies in the centre of the multiplier algebra of B , Φ: A → B is a Jordan epimorphism, D: A → X is a derivation and Ψ: A → X is a bimodule homomorphism.

Continuity of Lie derivations on Banach algebras

We prove that if A and B are semisimple Banachn algebras, then the separating subspace of every Lie isomorphism from A onto B is contained in the centren of B .

Continuity of Derivations, Intertwining Maps, and Cocycles from Banach Algebras

We prove that every Lie derivation on a symmetrically amenable semisimple Banach algebra can be uniquely decomposed into the sum of a derivation and a centre-valued trace.

Multiplicative-Semiprimeness of Nondegenerate Jordan Algebras

The Lie structure induced on a Banach algebra by the bracket [a, b] — ab — ba is of lively interest for their intimate connections with the geometry of manifolds modeled on Banach spaces. Many mathematics have studied Lie derivations on associative rings [1, 5] and Lie derivations on some Banach algebras [2, 7, 8]. A Lie derivation of a Banach algebra A is a linear map D from A into itself satisfying D([a, b]) = [D(a), b] + [a, D(b)] for all a,b e A. In this paper we study the continuity of a Lie derivation D on an arbitrary semisimple Banach algebra A. We measure the continuity of D by considering its separating subspace, which is defined as the subspace 5(D) of those elements a e A for which there is a sequence {an} in A satisfying liman = 0 and limD(an) = a. <S(D) is easily checked to be a Lie ideal of A and the closed graph theorem shows that D is continuous if, and only if, S(D) = 0. Until further notice we assume that A is a unital semisimple complex Banach algebra and D stands for a Lie derivation of A. Let Z(A) denote the centre of A. For each a e A let ada denote the continuous linear operator ada(b) = [a, b] from A into itself. If P is a closed ideal of A we will denote by QP the quotient map from A onto A/P. The next important result was essentially stated by M. P. Thomas and illustrate the typical sliding hump argument.

Functional Identities in Jordan Algebras: Associating Traces and Lie Triple Isomorphisms

Let A be a Banach algebra, and let E be a Banach A-bimodule. A linear map S:A→E is intertwining if the bilinear map Formula is continuous, and a linear map D:A→E is a derivation if δ1D=0, so that a derivation is an intertwining map. Derivations from A to E are not necessarily continuous. The purpose of the present paper is to prove that the continuity of all intertwining maps from a Banach algebra A into each Banach A-bimodule follows from the fact that all derivations from A into each such bimodule are continuous; this resolves a question left open in [1, p. 36]. Indeed, we prove a somewhat stronger result involving left- (or right-) intertwining maps.

FUNCTIONAL IDENTITIES IN JORDAN ALGEBRAS: ASSOCIATING MAPS

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