ON CONVERGENCE OF CONDITIONAL EXPECTATION OPERATORS

C. BRYAN DAWSON

Abstract:

Given an operator $T:U_X(\Sigma)\to Y$ or $T:C(H,X)\to Y$, one may consider the net of conditional expectation operators $(T_{\pi})$ directed by refinement of the partitions $\pi$. It has been shown previously that $(T_{\pi})$ does not always converge to T. This paper gives several conditions under which this convergence does occur, including complete characterizations when $X= {\bf R}$ or when X^* has the Radon-Nikodým property.