Metric versions of Herstein's theorems on Jordan maps

Abstract

Let A be a Banach algebra and let B be an ultraprime Banach algebra. If Φ: A → B is a surjective continuous linear map which tends to satisfy the Jordan multiplicativity condition, then we show that Φ comes near to satisfy either the multiplicativity or the anti-multiplicativity condition. In fact, we give a quantitative estimate of this phenomenon. Furthermore, we estimate how much a continuous linear map Δ: B → B approaches to satisfy the derivation identity in the case when Δ tends to satisfy the Jordan derivation identity

Keywords:

• homomorphism
• anti-homomorphism
• Jordan homomorphism
• derivation
• Jordan derivation
• ultraprime Banach algebra

AMS Subject Classifications::

• Primary: 17C65
• Secondary: 47B48

Acknolwedgements

The authors were supported by MICINN Grant MTM2009–07498 and Junta de Andalucía Grants FQM-185 and FQM-4911.