Additivity of Elementary Maps on Rings

Abstract

Let ℛ and ℛ′ be rings. Under some assumptions on ℛ, we study the additivity of maps M: ℛ → ℛ′ and M*: ℛ′ → ℛ that are surjective and satisfy M(x M*(y)z) = M(x)yM(z) and M*(yM(x)u) = M*(y)x M*(u) for x, z ∈ ℛ and y, u ∈ ℛ′. In particular, if ℛ is a prime ring containing a non-trivial idempotent, or a standard operator algebra, or a nest algebra on a Hilbert space, or an atomic Boolean subspace lattice algebra on a Banach space, then we can conclude that both M and M* are additive.

Key Words:

• Additivity
• Elementary maps
• Rings
• Operator algebras

AMS Subject Classification (2000):

• 16W20
• 47B47
• 47B49

Acknowledgment

The authors would like to thank the referee for many helpful comments.

Notes

^#Communicated by W. Martindale.