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A classification for 2-isometries of noncommutative Lp-spaces

Marius Junge·Zhong-Jin Ruan·David Sherman
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Abstract

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) to Lp(N) between arbitrary noncommutative Lp-spaces can always be written in the form T(phi^{1/p}) = w (phi circ pi^{-1} circ E)^{1/p}, for phi in M_*^+. Here pi is a normal *-isomorphism from M onto the von Neumann subalgebra pi(M) of N, w is a partial isometry in N, and E is a normal conditional expectation from N onto pi(M). As a consequence of this, any 2-isometry is automatically a complete isometry and has completely contractively complemented range.

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Marius Junge, Zhong-Jin Ruan, David Sherman. 2004-02-11. A classification for 2-isometries of noncommutative Lp-spaces. https://arxiv.org/abs/math/0402181

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