Weight on a von Neumann algebra

Let $M$ be a von Neumann algebra with positive cone $M_+$ . A weight on $M$ is a map

\varphi:M_+\longrightarrow[0,+\infty]

such that $\varphi(x+y)=\varphi(x)+\varphi(y)$ and $\varphi(\lambda x)=\lambda\varphi(x)$ for all $x,y\in M_+$ and $\lambda\geq0$ , with $0\cdot(+\infty)=0$ . No boundedness, normality, faithfulness, or semifiniteness is implicit. If $\varphi$ is finite on all of $M_+$ , it extends uniquely to a positive linear functional on $M$ ; otherwise it is genuinely extended-valued and is not a linear functional on all of $M$ .

Finite domains

Three domains organize the part of $M$ on which a weight behaves finitely:

\mathfrak m_\varphi^+=\{x\in M_+:\varphi(x)<\infty\},\qquad \mathfrak n_\varphi=\{a\in M:\varphi(a^*a)<\infty\},

and $\mathfrak m_\varphi=\operatorname{span}\mathfrak m_\varphi^+$ . The set $\mathfrak n_\varphi$ is a left ideal, while

\mathfrak m_\varphi =\operatorname{span}\{y^*x:x,y\in\mathfrak n_\varphi\}

is a $*$ -subalgebra on which $\varphi$ has a linear extension. This extension need not be bounded or defined on all of $M$ .

Regularity properties

Faithfulness, normality, and semifiniteness impose independent conditions. A faithful weight detects every nonzero positive element. A normal weight preserves suprema of increasing nets in $M_+$ . A semifinite weight has enough finite-weight positive elements to be order-dense in $M_+$ . A weight having all three properties is called faithful, normal, and semifinite, often abbreviated FNS; this is a hypothesis, not part of the word “weight.”

Examples and scope

Every positive linear functional on $M$ restricts to a finite weight on $M_+$ . On $B(H)$ , the canonical operator trace is a weight that may take the value $+\infty$ ; it is faithful, normal, and semifinite. By contrast, assigning $0$ to $0$ and $+\infty$ to every nonzero positive element is a weight, but it is neither semifinite nor useful as a linear functional. Weights are defined on positive elements so that extended values can be added without trying to form undefined expressions such as $+\infty-(+\infty)$ .

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter VII, §1 on weights and their finite domains.
Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: the chapters on densely defined weights and positive functionals.