Abstract. In this paper the relation between the elements of a von Neumann algebra for which a suitable $L^2$ type norm is preserved by an $\omega $-conditional expectation to a von Neumann subalgebra and the relevant canonical state extension is discussed. The main tool used is a polar decomposition for operators depending on both the von Neumann algebra and a faithful normal state on it.

AMS Subject Classification (1991): 46L50

Received June 7, 1993 and in revised form December 28, 1993. (Registered under 5576/2009.)