Faithful weight - Imported Knowlpedia Content

Definition

Let $M$ be a von Neumann algebra and let $\varphi:M_+\to[0,+\infty]$ be a weight. The weight $\varphi$ is faithful if

x\in M_+,\quad \varphi(x)=0\quad\Longrightarrow\quad x=0.

Equivalently, $\varphi(a^*a)=0$ implies $a=0$ for every $a\in M$ . Faithfulness is a nondegeneracy condition only: it neither requires $\varphi(x)<\infty$ for nonzero $x$ nor asserts normality or semifiniteness. In particular, an extended-valued weight can be faithful even when its finite domain is small. The ambient positive cone is essential because a weight is not generally a complex-valued functional on all of $M$ .

Null ideal

The set

\mathfrak n_\varphi^0=\{a\in M:\varphi(a^*a)=0\}

is the null left ideal used when constructing the Hilbert space associated with a weight. Faithfulness is exactly the assertion $\mathfrak n_\varphi^0=\{0\}$ . For a nonfaithful weight, quotienting by this ideal removes directions invisible to $\varphi$ , just as the null space is removed in the GNS construction for a positive functional Takesaki, vol. I, Chapter VII, §1.

Support

For a normal weight, there is a support projection $s(\varphi)$ such that $\varphi$ is faithful on the corner $s(\varphi)Ms(\varphi)$ . Faithfulness is equivalent to $s(\varphi)=1$ . This support formulation depends on normality; it should not be used as the definition for an arbitrary weight without first establishing that the relevant support projection exists.

Examples and independence

The canonical operator trace on $B(H)$ is faithful: a positive operator with zero trace is zero. A vector state on $B(H)$ is generally not faithful, because any nonzero positive operator annihilating the chosen vector has value zero. The weight that is $0$ at $0$ and $+\infty$ on every nonzero positive element is faithful but not semifinite, demonstrating that the two conditions are logically independent.

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter VII, §1 on faithful weights and their null ideals.
Masamichi Takesaki, Theory of Operator Algebras II, Springer, 2003. DOI record. Relevant: the opening chapters on faithful normal semifinite weights and support.