The goal of this talk is to give an introduction to Haagerup's 
construction of an $L^p$ space associated with a von Neumann algebra.
Some background: Separable commutative von Neumann algebras are 
isomorphic to $L^\infty(X,\mu)$ for some standard measure space, and to 
such a space one associates the $L^p$ spaces in the usual sense. For a 
semifinite von Neumann algebra $M$ with faithful normal semifinite trace 
$\tau$, Dixmier, Segal and Kunze introduced a space $L^p(M,\tau)$, 
generalizing the classical ones. The extension by Haagerup applies to 
arbitrary (not necessary semifinite) von Neumann algebras and for 
semifinite ones it is isometrically isomorphic to $L^p(M,\tau)$ for any 
faithful normal semifinite trace $\tau$.