A classification for 2-isometries of noncommutative L_P-spaces

In this paper we extend previous results of Banach, Lamperti and Yeadon on isometries of L_p-spaces to the non-tracial case first introduced by Haagerup. Specifically, we use operator space techniques and an extrapolation argument to prove that every 2-isometry T: L_p(M) → L _P(N) between arbitrary noncommutative L_p-spaces can always be written in the form T(φ1/p) = w(φ o π^-1 o E)1/p, φ∈ M*⁺. Here π is a normal *-isomorphism from M onto the von Neumann subalgebra π(M) of N, w is a partial isometry in N, and E is a normal conditional expectation from N onto π(M). As a consequence of this, any 2-isometry is automatically a complete isometry and has completely contractively complemented range.