If $A$ is a Von Neuman algebra there is something we can called its $L^2$ representation: it is the GNS representation attached to any faithful normal state.

The point is that the $L^2$ representation does not (up to canonical isomorphism) depends on the choice of the faithfuls normal state. In order to obtain a construction of the $L^2$ representation which clearly does not depends on any choice, we used this technique: We take (for example, there is a lot of possible variation) the set of all couple $(\eta,h)$ where $\eta$ is a faithfull normal state and $h$ is a vector in the representation attached to $\eta$, and putt a scalar product on this (using the canonical identification of two GNS representation) and this generate the $L^2$ representation.