Non-linear numerical radius isometries on atomic nest algebras and diagonal algebras

A nonlinear map φ between operator algebras is said to be a numerical radius isometry if w(φ(T−S))=w(T−S) for all T, S in its domain algebra, where w(T) stands for the numerical radius of T. Let N and M be two atomic nests on complex Hilbert spaces H and K, respectively. Denote Alg N the nest algebra associated with N and DN=Alg N∩(Alg N)∗ the diagonal algebra. We give a thorough classification of weakly continuous numerical radius isometries from Alg N onto Alg M and a thorough classification of numerical radius isometries from DN onto DM.