One of the basic tools in subfactors is the conditional expectation. If $N\subset M$ is a $II_1$-subfactor (or an inclusion of finite factors), then there is a unique trace-preserving conditional expectation of $M$ onto $N$. This should be thought of as a (Banach space) projection of norm 1. In fact, it is the restriction of the Jones projection $e_N$ on $L^2(M)$ to $M$. In the finite index case, we get another conditional expectation (Jones projection...) from the basic construction $M_1=\langle M, e_N\rangle$ onto $M$.

In his thesis, Michael Burns showed that if we iterate the basic construction in the infinite index case, we only get half the conditional expectations (we only get the odd Jones projections). The other half of the time, we get a generalization of the conditional expectation called an operator valued weight, originally defined by Haagerup.

Given an inclusion of semifinite von Neumann algebras $(N, tr_N)\subset (M, tr_M)$, there is a unique normal, faithful, semi-finite trace-preserving operator valued weight $T\colon M_+\to \widehat{N_+}$, where we must take the "extended part" of the positive cone $N_+$ of $N$.

Edit as per @Dmitri's answer: Let $$ n_T=\{x\in M| T(x^\ast x)\in N_+\} $$ and set $$ m_T=n_T^\ast n_T=span\{x^\ast y| x,y\in n_T\}. $$ There is a natural extension of $T$ to $m_T$. Is there an example of a normal, faithful, semifinite operator valued weight such that

- $N$ is not contained in $T(m_T)$, and/or
- $1\notin T(M_+)$?

What about when $M$ and $N$ are factors ($M$ is $II_\infty$)?