Left ideal of a weight

Let $\varphi$ be a weight on a von Neumann algebra $M$ . The left ideal of $\varphi$ is

\mathfrak n_\varphi =\{x\in M:\varphi(x^*x)<\infty\}.

It is a complex vector subspace and a left ideal: if $a\in M$ and $x\in\mathfrak n_\varphi$ , then

(ax)^*(ax)\leq\|a\|^2x^*x,

so $ax\in\mathfrak n_\varphi$ . It need not be self-adjoint or two-sided. The terminology “square-integrable” reflects the commutative model, where $\varphi$ is integration and the condition says that $x$ has finite $L^2$ -norm.

Associated finite algebra

The products $y^*x$ , with $x,y\in\mathfrak n_\varphi$ , form the linear domain

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

The weight has a linear extension to this $*$ -algebra. The Cauchy–Schwarz inequality for weights ensures that $\varphi(y^*x)$ is finite on such products.

Examples and non-examples

For the usual operator trace on $\mathcal B(H)$ , $\mathfrak n_\varphi$ is the ideal of Hilbert–Schmidt operators, while $\mathfrak m_\varphi$ is the trace-class ideal. If $\varphi$ is a bounded positive functional, then $\mathfrak n_\varphi=M$ . By contrast, the finite positive domain $\{x\in M_+:\varphi(x)<\infty\}$ contains only positive elements and is not the left ideal used in the GNS construction for the weight.

Role in the weight GNS construction

The formula

\langle x,y\rangle_\varphi=\varphi(y^*x)

defines a positive semidefinite inner product on $\mathfrak n_\varphi$ . After quotienting by its null space and completing, left multiplication by $M$ gives the GNS representation of the weight. Thus the left-ideal property is precisely what makes the representation action well defined.

References

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