Abstract
Let M and N be arbitrary von Neumann algebras. For any a in M or in N, let Delta(lambda) (a) denote the lambda-Aluthge transform of a. Suppose that M has no abelian direct summand. We prove that every bijective map Phi : M -> N satisfying
Phi(Delta(lambda) (a o b*) = Delta(lambda) (Phi(a) o Phi(b)*), for all a, b is an element of M,
(for a fixed lambda is an element of [0, 1]), maps the hermitian part of M onto the hermitian part of N (i.e. .Phi(M-sa) = N-sa) and its restriction Phi vertical bar M-sa : M-sa -> N-sa is a Jordan isomorphism. If we also assume that Phi(x + iy) = Phi(x) + Phi(iy) for all x, y is an element of M-sa, then there exists a central projection p(c) in M such that Phi vertical bar(pcM) is a complex linear Jordan *-isomorphism and Phi vertical bar((1 - pc)M) is a conjugate linear Jordan *-isomorphism.
Given two complex Hilbert spaces H and K with dim(H) >= 2, we also show that every bijection Phi : B(H) -> B(K) satisfying
Phi(Delta(lambda) (ab*) = Delta(lambda) (Phi(a) Phi(b)*), for all a, b is an element of M,
must be a complex linear or a conjugate linear *-isomorphism. (C) 2018 Elsevier Inc. All rights reserved.