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A classification for 2-isometries of noncommutativeL p-spaces

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Abstract

In this paper we extend previous results of Banach, Lamperti and Yeadon on isometries ofL 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-isometryT:L p(M) →L p(N) between arbitrary noncommutativeL p-spaces can always be written in the form\(T(\phi ^{\frac{1}{p}} ) = w(\phi o \pi ^{ - 1} o {\rm E})^{\frac{1}{p}} , \phi \in \mathcal{M}_ * ^ + \) Here π is a normal *-isomorphism fromM onto the von Neumann subalgebra π(M) ofN,w is a partial isometry inN, andE is a normal conditional expectation fromN 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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The first and second authors were partially supported by the National Science Foundation DMS-0088928 and DMS-0140067.

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Junge, M., Ruan, ZJ. & Sherman, D. A classification for 2-isometries of noncommutativeL p-spaces. Isr. J. Math. 150, 285–314 (2005). https://doi.org/10.1007/BF02762384

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