2-local isometries of non-commutative Lorentz spaces

Let M be a von Neumann algebra equipped with a faithful normal finite trace τ, and let S(M,τ) be an ∗-algebra of all τ-measurable operators affiliated with M. For x∈S(M,τ) the generalized singular value function μ(x):t→μ(t;x), t>0, is defined by the equality μ(t;x)=inf{∥xp∥M:p2=p∗=p∈M,τ(1-p)≤t}. Let ψ be an increasing concave continuous function on [0,∞) with ψ(0)=0, ψ(∞)=∞, and let Λψ(M,τ)={x∈S(M,τ): ∥x∥ψ=∫∞0μ(t;x)dψ(t)