Triple Homomorphisms Research Articles

On the strict topology of the multipliers of a JB^*-algebra

We introduce the Jordan-strict topology on the multiplier algebra of a JB^*-algebra, a notion which was missing despite the forty years passed after the first studies on Jordan multipliers. In case that a C^*-algebra A is regarded as a JB^*-algebra, the J-strict topology of M(A) is precisely the well-studied C^*-strict topology. We prove that every JB^*-algebra {mathfrak {A}} is J-strict dense in its multiplier algebra M({mathfrak {A}}), and that latter algebra is J-strict complete. We show that continuous surjective Jordan homomorphisms, triple homomorphisms, and orthogonality preserving operators between JB^*-algebras admit J-strict continuous extensions to the corresponding type of operators between the multiplier algebras. We characterize J-strict continuous functionals on the multiplier algebra of a JB^*-algebra {mathfrak {A}}, and we establish that the dual of M({mathfrak {A}}) with respect to the J-strict topology is isometrically isomorphic to {mathfrak {A}}^*. We also present a first application of the J-strict topology of the multiplier algebra, by showing that under the extra hypothesis that {mathfrak {A}} and {mathfrak {B}} are sigma -unital JB^*-algebras, every surjective Jordan ^*-homomorphism (respectively, triple homomorphism or continuous orthogonality preserving operator) from {mathfrak {A}} onto {mathfrak {B}} admits an extension to a surjective J-strict continuous Jordan ^*-homomorphism (respectively, triple homomorphism or continuous orthogonality preserving operator) from M({mathfrak {A}}) onto M({mathfrak {B}}).

Conformal Triple Derivations and Triple Homomorphisms of Lie Conformal Algebras

Let [Formula: see text] be a finite Lie conformal algebra. We investigate the conformal derivation algebra [Formula: see text], conformal triple derivation algebra [Formula: see text] and generalized conformal triple derivation algebra [Formula: see text], focusing mainly on the connections among these derivation algebras. We also give a complete classification of (generalized) conformal triple derivation algebras on all finite simple Lie conformal algebras. In particular, [Formula: see text], where [Formula: see text] is a finite simple Lie conformal algebra. But for [Formula: see text], we obtain a conclusion that is closely related to [Formula: see text]. Finally, we introduce the definition of a triple homomorphism of Lie conformal algebras. Triple homomorphisms of all finite simple Lie conformal algebras are also characterized.

On triple homomorphisms of Lie algebras

Let [Formula: see text] and [Formula: see text] be two Lie algebras over a commutative ring with identity. In this paper, under some conditions on [Formula: see text] and [Formula: see text], it is proved that every triple homomorphism from [Formula: see text] onto [Formula: see text] is the sum of a homomorphism and an antihomomorphism from [Formula: see text] into [Formula: see text]. We also show that a finite-dimensional Lie algebra [Formula: see text] over an algebraically closed field of characteristic zero is nilpotent of class at most [Formula: see text] if and only if the sum of every homomorphism and every antihomomorphism on [Formula: see text] is a triple homomorphism.

Jordan triple homomorphisms on {mathcal {T}}_infty (F)

Let {mathcal {T}}_infty (F) be the algebra of all {mathbb {N}}times {mathbb {N}} upper triangular matrices defined over a field F of characteristic different from 2. We consider the Jordan triple homomorphisms of {mathcal {T}}_infty (F), i.e. the additive maps that satisfy the condition phi (xyx)=phi (x)phi (y)phi (x) for all x,yin {mathcal {T}}_infty (F). For the case when F is a prime field we find the form of all such maps phi . For the general case we present the form of the surjective maps phi .

Complete orthogonality preservers between C⁎-algebras

We introduce n-orthogonality (and completely orthogonality) preserving operators between C⁎-algebras. Our main theorem states that every completely orthogonality preserving bounded linear mapping between C⁎-algebras is a weighted TRO homomorphism. We also give several characterisations of TRO homomorphisms among triple homomorphisms.

On decompositions of ternary rings of operators and their morphisms

We introduce right, left, square and square-free W*-TROs and show that a W*-TRO has a canonical decomposition in terms of these. We examine ternary involutions as an application, and investigate decomposition of triple homomorphisms between TROs.

The structure of triple homomorphisms onto prime algebras

AbstractTriple homomorphisms on C*-algebras and JB*-triples have been studied in the literature. From the viewpoint of associative algebras, we characterise the structure of triple homomorphisms from an arbitrary ⋆-algebra onto a prime *-algebra. As an application, we prove that every triple homomorphism from a Banach ⋆-algebra onto a prime semisimple idempotent Banach *-algebra is continuous. The analogous results for prime C*-algebras and standard operator *-algebras on Hilbert spaces are also described.

Pointwise-generalized-inverses of linear maps between C^*-algebras and JB^*-triples

We study pointwise-generalized-inverses of linear maps between C$^*$-algebras. Let $\Phi$ and $\Psi$ be linear maps between complex Banach algebras $A$ and $B$. We say that $\Psi$ is a pointwise-generalized-inverse of $\Phi$ if $\Phi(aba)=\Phi(a)\Psi(b)\Phi(a),$ for every $a,b\in A$. The pair $(\Phi,\Psi)$ is Jordan-triple multiplicative if $\Phi$ is a pointwise-generalized-inverse of $\Psi$ and the latter is a pointwise-generalized-inverse of $\Phi$. We study the basic properties of this maps in connection with Jordan homomorphism, triple homomorphisms and strongly preservers. We also determine conditions to guarantee the automatic continuity of the pointwise-generalized-inverse of continuous operator between C$^*$-algebras. An appropriate generalization is introduced in the setting of JB$^*$-triples.

Linear maps on C⁎-algebras which are derivations or triple derivations at a point

We prove new results about generalized derivations on C⁎-algebras. By considering the triple product {a,b,c}=2−1(ab⁎c+cb⁎a), we introduce the study of linear maps which are triple derivations or triple homomorphisms at a point. We prove that a continuous linear map T on a unital C⁎-algebra is a generalized derivation whenever it is a triple derivation at the unit element. If we additionally assume T(1)=0, then T is a ⁎-derivation and a triple derivation. Furthermore, a continuous linear map on a unital C⁎-algebra which is a triple derivation at the unit element is a triple derivation. Similar conclusions are obtained for continuous linear maps which are derivations or triple derivations at zero.We also give an automatic continuity result, that is, we show that generalized derivations on a von Neumann algebra and linear maps on a von Neumann algebra which are derivations or triple derivations at zero are all continuous even if not assumed a priori to be so.

Triple derivations and triple homomorphisms of perfect Lie superalgebras

In this paper, we study triple derivations and triple homomorphisms of perfect Lie superalgebras over a commutative ring R. It is proved that, if the base ring contains 12, L is a perfect Lie superalgebra with zero center, then every triple derivation of L is a derivation, and every triple derivation of the derivation algebra Der(L) is an inner derivation. Let L,L′ be Lie superalgebras over a commutative ring R, the notion of triple homomorphism from L to L′ is introduced. We prove that, under certain assumptions, homomorphisms, anti-homomorphisms, and sums of homomorphisms and anti-homomorphisms are all triple homomorphisms.

Jordan triple product homomorphisms on Hermitian matrices to and from dimension one

We characterize all Jordan triple product homomorphisms, that is, mappings satisfyingfrom the set of all Hermitian complex matrices to the field of complex numbers. Further, we characterize all Jordan triple product homomorphisms from the field of complex or real numbers or the set of all nonnegative real numbers to the set of all Hermitian complex matrices.

2-local triple homomorphisms on von Neumann algebras and JBW⁎-triples

We prove that every (not necessarily linear nor continuous) 2-local triple homomorphism from a JBW⁎-triple into a JB⁎-triple is linear and a triple homomorphism. Consequently, every 2-local triple homomorphism from a von Neumann algebra (respectively, from a JBW⁎-algebra) into a C⁎-algebra (respectively, into a JB⁎-algebra) is linear and a triple homomorphism.

Triple Homomorphisms of Perfect Lie Algebras

Let L, L′ be Lie algebras over a commutative ring R. A R-linear mapping f: L → L′ is called a triple homomorphism from L to L′ if f([x, [y, z]]) = [f(x), [f(y), f(z)]] for all x, y, z ∈ L. It is clear that homomorphisms, anti-homomorphisms, and sums of homomorphisms and anti-homomorphisms are all triple homomorphisms. We proved that, under certain assumptions, these are all triple homomorphisms.

Generalized Triple Homomorphisms and Derivations

AbstractWe introduce generalized triple homomorphisms between Jordan–Banach triple systems as a concept that extends the notion of generalized homomorphisms between Banach algebras given by K. Jarosz and B. E. Johnson in 1985 and 1987, respectively. We prove that every generalized triple homomorphism between JB*-triples is automatically continuous. When particularized to C*-algebras, we rediscover one of the main theorems established by Johnson. We will also consider generalized triple derivations from a Jordan–Banach triple Einto a Jordan–Banach triple E-module, proving that every generalized triple derivation from a JB*triple Einto itself or into E*is automatically continuous.

Universally reversible JC *-triples and operator spaces

We prove that the vast majority of JC*-triples satisfy the condition of universal reversibility. Our characterisation is that a JC*-triple is universally reversible if and only if it has no triple homomorphisms onto Hilbert spaces of dimension greater than two nor onto spin factors of dimension greater than four. We establish corresponding characterisations in the cases of JW*-triples and of TROs (regarded as JC*-triples). We show that the distinct natural operator space structures on a universally reversible JC*-triple E are in bijective correspondence with a distinguished class of ideals in its universal TRO, identify the Shilov boundaries of these operator spaces and prove that E has a unique natural operator space structure precisely when E contains no ideal isometric to a nonabelian TRO. We deduce some decomposition and completely contractive properties of triple homomorphisms on TROs.

C*-TERNARY HOMOMORPHISMS, C*-TERNARY DERIVATIONS, JB*-TRIPLE HOMOMORPHISMS AND JB*-TRIPLE DERIVATIONS

Park and Rassias proved the superstability of C*-ternary homomorphisms, C*-ternary derivations, JB*-triple homomorphisms and JB*-triple derivations, associated with the following Apollonius type additive functional equation [Formula: see text] by using direct method. Under the conditions of the theorems, we can show that the mappings f must be zero. In this paper, we correct the conditions. Furthermore, we prove the superstability of C*-ternary homomorphisms, C*-ternary derivations, JB*-triple homomorphisms and JB*-triple derivations by using fixed point method.

Weakly compact orthogonality preservers on C*-algebras

We establish a complete description of weakly compact orthogonality preserving operators from a C*-algebra A (or a JB*-algebra) to a JB*-triple E. We prove that any such operator is the sum of a series of mutually orthogonal triple homomorphisms from A** to a certain Peirce-2 subspace of E** multiplied by a mutually orthogonal norm-null sequence in E which enjoy certain operator commutativity relations.

Jordan mappings of semiprime rings II

We describe Jordan homomorphisms and Jordan triple homomorphisms onto 2-torsion free semiprime rings in which the annihilator of any ideal is a direct summand.

Jordan triple homomorphisms of associative rings with involution

Jordan triple homomorphisms of associative rings with involution

Jordan triple homomorphisms of associative structures

We show that Jordan triple homomorphisms and derivations between prime special quadral Jordan triple systems on which Zel’manov polynomials do not vanish extend to associative homomorphis and derivations of associative ∗-envelopes (either associative triple systems or Z2-graded associative algebras). This generalizes results of Zel'manov and McCrimmon for Jordan algebras (which in turn generalized results of Martindale).