Linear bijections on von Neumann factors commuting with λ-Aluthge transform

We prove that a bijective linear transformation between von Neumann factors that commute with a lambda-Aluthge transform is necessarily a non-zero scalar multiple of an algebra *-isomorphism in the case of algebras that are not of type I-2. As for type I-2 factors, that is, in the particular case of the algebra of 2 by 2 complex matrices, we also present a complete description of those transformations which is a bit different. Namely, non-zero scalar multiples of the sum of an algebra *-antiisomorphism and the negative of the trace functional times the identity also show up.