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Journal of Operator Theory

Volume 53, Issue 1, Winter 2005  pp. 3-34.

Noncommutative $L^p$ modules

Authors Marius Junge (1) and David Sherman (2)
Author institution: (1) Department of Mathematics, University of Illinois, Urbana, IL 61801, USA
(2) Department of Mathematics, University of California, Santa Barbara, CA 93106, USA

Summary:  We construct classes of von Neumann algebra modules by considering `column sums' of noncommutative $L^p$ spaces. Our abstract characterization is based on an $L^{p/2}$-valued inner product, thereby generalizing Hilbert $C^*$-modules and representations on Hilbert space. While the (single) representation theory is similar to the $L^2$ case, the concept of $L^p$ bimodule ($p \ne 2$) turns out to be nearly trivial.

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