NORM OF POSITIVE SUM PRESERVERS OF NONCOMMUTATIVE Lp(M) SPACES

作者：Zhang, Jun; Tsai, Ming-Cheng^*; Wong, Ngai-Ching

来源：Journal of Nonlinear and Convex Analysis, 2021, 22(2): 251-264.

Let 1 < p < +infinity. Let L-P(M) and L-P(N) be the noncommutative L-P-spaces associated to von Neumann algebras M and N, respectively. Let phi:L-+(P)(M) -> L-+(P)(N) be a surjective map between positive elements preserving the norm of sums, i.e., @@@ parallel to phi(x) + phi(Y)parallel to(p) = parallel to x+y parallel to(p,) xy is an element of L-+(p)(M). @@@ We show that there is a Jordan *-isomorphism J : M -> N, and phi can be extended uniquely to a surjective real linear positive isometry from L-sa(p)(M) onto L-sa(p)(N). When M is approximately semifinite, especially semifinite or hyperfinite, phi(R) = Theta(*)(R-p)(1/p) for every R is an element of L-+(P)(M), where Theta = J(-1) and Theta(*): L-l(M)(congruent to M-*) -> L-1(N)(congruent to N-*) is its predual map. In the case when M has a normal faithful semifinite trace tau(M) (and so does N), phi(x) = hJ(x) for every x is an element of L-P(M, tau(M))boolean AND M+, where h(p) = d(tau(M) circle Theta)/d tau(N) is the non-commutative Radon-Nikodym derivative of tau(M) circle Theta with respect to tau(N). We also provide a similar result when p = +infinity, and counter examples for the case p = 1.

• 单位
  中山大学