2-local isometries of non-commutative Lorentz spaces

Abstract

Let mathcal M be a von Neumann algebra equipped with a faithful normal finite trace tau, and let Sleft( mathcalM, tauright) be an ast -algebra of all tau -measurable operators affiliated with mathcal M . For x in Sleft( mathcalM, tauright) the generalized singular value function mu(x):trightarrow mu(tx), t0, is defined by the equality mu(tx)infxp_mathcalM:, p2pp in mathcalM, , tau(mathbf1-p)leq t. Let psi be an increasing concave continuous function on 0, infty) with psi(0) 0, psi(infty)infty, and let Lambda_psi(mathcal M,tau) left x in Sleft( mathcalM, tauright): x _psi int_0inftymu(tx)dpsi(t) infty right be the non-commutative Lorentz space. A surjective (not necessarily linear) mapping V:, Lambda_psi(mathcal M,tau) to Lambda_psi(mathcal M,tau) is called a surjective 2-local isometry, if for any x, y in Lambda_psi(mathcal M,tau) there exists a surjective linear isometry V_x, y:, Lambda_psi(mathcal M,tau) to Lambda_psi(mathcal M,tau) such that V(x) V_x, y(x) and V(y) V_x, y(y). It is proved that in the case when mathcalM is a factor, every surjective 2-local isometry V:Lambda_psi(mathcal M,tau) to Lambda_psi(mathcal M,tau) is a linear isometry.