Nonlinear Strong Commutativity Preserving Maps on Prime Rings

Abstract

Let 𝒜 be a unital prime ring containing a nontrivial idempotent P. Assume that Φ: 𝒜 → 𝒜 is a nonlinear surjective map. It is shown that Φ preserves strong commutativity if and only if Φ has the form Φ(A) = αA + f(A) for all A ∈ 𝒜, where α ∈ {1, −1} and f is a map from 𝒜 into 𝒵(𝒜). As an application, a characterization of nonlinear surjective strong commutativity preserving maps on factor von Neumann algebras is obtained.

Key Words:

• Factor von Neumann algebras
• Nonlinear strong commutativity preserving maps
• Prime rings

2000 Mathematics Subject Classification:

• 16N60
• 16U80

ACKNOWLEDGMENT

The authors wish to give their thanks to the referee for reading the original manuscript carefully and giving many helpful comments to improve the article.

Notes

Communicated by M. Bresar.