[Paper Review] Jordan Triple Elementary Maps on Rings

Wu Jing|ArXiv.org|Jun 4, 2007

Advanced Topics in Algebra5 references3 citations

TL;DR

This paper establishes that Jordan triple elementary maps between unital rings containing a nontrivial idempotent are automatically additive if the rings are 2-torsion free and satisfy a certain ideal condition (P). The key result proves that surjective Jordan triple elementary maps $M: olinebreak olinebreak \mathcal{R} \to \mathcal{R}'$ and $M^*: \mathcal{R}' \to \mathcal{R}$ are additive under these conditions, extending earlier results on multiplicative maps and providing a foundation for additivity in non-associative ring maps.

ABSTRACT

We prove that Jordan triple elementary surjective maps on unital rings containing a nontrivial idempotent are automatically additive.

Motivation & Objective

• To investigate the additivity of Jordan triple elementary maps on rings, extending known results on multiplicative maps.
• To establish conditions under which surjective Jordan triple elementary maps on rings are necessarily additive.
• To generalize Martindale's 1969 result on multiplicative bijective maps to the Jordan triple elementary setting.
• To apply the results to prime rings and standard operator algebras, showing additivity in broader algebraic contexts.

Proposed method

• Define Jordan triple elementary maps via a pair of maps $ (M, M^*) $ satisfying a system of functional equations involving triple products.
• Use the decomposition $ \mathcal{R} = \mathcal{R}_{11} \oplus \mathcal{R}_{12} \oplus \mathcal{R}_{21} \oplus \mathcal{R}_{22} $ induced by a nontrivial idempotent $ e_1 $, with $ e_2 = 1 - e_1 $.
• Prove that $ M $ and $ M^* $ are injective and preserve zero via direct computation using the defining equations.
• Establish additivity by showing $ M(a + b) = M(a) + M(b) $ and $ M^*(x + y) = M^*(x) + M^*(y) $ through component-wise analysis in the $ e_i $-decomposition.
• Use the condition (P): $ e_i a e_j \mathcal{R} e_k = \{0\} $ or $ e_k \mathcal{R} e_i a e_j = \{0\} $ implies $ e_i a e_j = 0 $, to deduce component-wise equalities.
• Leverage the 2-torsion free assumption to avoid issues in cancellation and ensure injectivity and additivity.

Experimental results

Research questions

• RQ1 Under what conditions is a Jordan triple elementary map on a ring necessarily additive?
• RQ2 Can the additivity of such maps be guaranteed without assuming linearity a priori?
• RQ3 Does the presence of a nontrivial idempotent and the 2-torsion free condition suffice to force additivity in Jordan triple elementary maps?
• RQ4 How does the structure of the ring decomposition via idempotents facilitate the proof of additivity?
• RQ5 Can the results be extended to prime rings and standard operator algebras?

Key findings

• The map $ M $ is additive: $ M(a + b) = M(a) + M(b) $ for all $ a, b \in \mathcal{R} $, under the stated conditions.
• The dual map $ M^* $ is additive: $ M^*(x + y) = M^*(x) + M^*(y) $ for all $ x, y \in \mathcal{R}' $, under the same conditions.
• The maps $ M $ and $ M^* $ are injective, as shown by assuming $ M(a) = M(b) $ and deriving $ a = b $ via the functional equations.
• The condition (P) ensures that zero in certain corner components implies the entire component is zero, which is essential for component-wise identification.
• The result holds for prime rings: if $ \mathcal{R} $ is a 2-torsion free unital prime ring with a nontrivial idempotent, then $ M $ and $ M^* $ are additive.
• The result extends to standard operator algebras on Banach spaces of dimension > 1, where $ M $ and $ M^* $ are also additive under the same functional equations.