Operator-valued weight - Imported Knowlpedia Content

Definition

Let $N\subseteq M$ be unital von Neumann algebras. An operator-valued weight from $M$ to $N$ is a map

T:M_+\longrightarrow\widehat N_+

into the extended positive cone that is additive, positively homogeneous, and $N$ -bimodular in the positive sense:

T(a^*xa)=a^*T(x)a \qquad(x\in M_+,\ a\in N).

The operations on the right are those of $\widehat N_+$ . Unlike a conditional expectation, $T(x)$ may be unbounded or infinite. No normality, faithfulness, or semifiniteness is implicit unless it is stated separately.

Regularity conditions

The weight $T$ is normal when it preserves suprema of increasing nets in $M_+$ , and faithful when $T(x)=0$ implies $x=0$ . It is semifinite when

\mathfrak n_T=\{x\in M:T(x^*x)\in N_+\}

is ultraweakly dense in $M$ . If $T$ is normal, faithful, and semifinite and $\varphi$ is a normal semifinite faithful scalar weight on $N$ , then $\varphi\circ T$ is such a weight on $M$ Haagerup, §2.

Conditional expectations

A conditional expectation $E:M\to N$ restricts to an operator-valued weight whose values lie in the bounded cone $N_+$ and satisfy $E(1)=1$ . Thus operator-valued weights generalize conditional expectations by allowing extended positive values. A merely positive map $M\to N$ need not be an operator-valued weight: it can fail the displayed $N$ -bimodularity identity.

Composition

The composition $\varphi\circ T$ is defined intrinsically because each scalar weight $\varphi$ extends from $N_+$ to $\widehat N_+$ by monotone limits. This construction allows information about scalar weights on $M$ to be transferred through an inclusion $N\subseteq M$ , and is one of the main uses of operator-valued weights in modular theory.

References

Uffe Haagerup, “Operator-Valued Weights in von Neumann Algebras I,” Journal of Functional Analysis 32 (1979), 175–206. DOI record. Relevant: §§1–2 on the definition, regularity properties, and composition with scalar weights.
Masamichi Takesaki, Theory of Operator Algebras II, Springer, 2003. DOI record. Relevant: Chapter IX, §4 on conditional expectations and operator-valued weights.