  We give an overview of relative tensor products (RTPs) for von Neumann
algebra modules. For background, we start with the categorical definition and
go on to examine its algebraic formulation, which is applied to Morita
equivalence and index. Then we consider the analytic construction, with
particular emphasis on explaining why the RTP is not generally defined for
every pair of vectors. We also look at recent work justifying a representation
of RTPs as composition of unbounded operators, noting that these ideas work
equally well for L^p modules. Finally, we prove some new results characterizing
preclosedness of the map (\xi, \eta) \mapsto \xi \otimes_\phi \eta.

Operator Algebras