In this paper we extend previous results of Banach, Lamperti and Yeadon on isometries of Lp-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: Lp(M) → L P(N) between arbitrary noncommutative Lp-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.
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