GNS construction for a weight - Imported Knowlpedia Content

Definition

Let $\varphi$ be a weight on a von Neumann algebra $M$ , and let $\mathfrak n_\varphi$ be its finite left ideal. On $\mathfrak n_\varphi$ set

\langle x,y\rangle_\varphi=\varphi(y^*x),\qquad \mathcal N_\varphi=\{x:\varphi(x^*x)=0\}.

Here $\varphi(y^*x)$ denotes the canonical polarized linear extension of $\varphi$ to $\mathfrak m_\varphi=\operatorname{span}(\mathfrak n_\varphi^* \mathfrak n_\varphi)$ . The GNS construction for $\varphi$ is the Hilbert-space completion $H_\varphi$ of $\mathfrak n_\varphi/\mathcal N_\varphi$ , with quotient map $\Lambda_\varphi$ , together with the representation

\pi_\varphi(a)\Lambda_\varphi(x)=\Lambda_\varphi(ax).

The inequality $\varphi(x^*a^*ax)\leq\|a\|^2\varphi(x^*x)$ makes $\pi_\varphi(a)$ bounded and the formula well defined.

Regularity of the representation

If $\varphi$ is normal, semifinite, and faithful, then $\pi_\varphi$ is a faithful normal representation. Semifiniteness ensures that the finite domain is sufficiently large, faithfulness removes nonzero positive elements from the kernel, and normality supplies ultraweak continuity Takesaki, Chapter VII, §1. Without these hypotheses the construction still exists, but the representation may be degenerate or nonfaithful.

Relation to the ordinary GNS construction

When $\varphi$ is a state, $\mathfrak n_\varphi=M$ , and the construction reduces to the usual GNS construction. For the canonical trace on $\mathcal B(H)$ , the resulting Hilbert space is the Hilbert–Schmidt class and $\pi_\varphi$ acts by left multiplication. The weight construction therefore retains the familiar state case while allowing genuinely unbounded noncommutative integrals.

Modular role

For a normal semifinite faithful weight, the dense vectors represented by finite products support the Tomita operator. Its polar decomposition yields the modular conjugation and modular operator, from which the modular automorphism group of $\varphi$ is obtained. The weight GNS construction is therefore the Hilbert-space entry point to Tomita–Takesaki theory.

References

Masamichi Takesaki, Theory of Operator Algebras II, Springer, 2003. DOI record. Relevant: Chapter VII, §1 on weights and semi-cyclic representations, and §2 on their associated left Hilbert algebras.