Centralizer of a weight - Imported Knowlpedia Content

Definition

Let $M$ be a von Neumann algebra and let $\varphi$ be a normal semifinite faithful weight. The centralizer of $\varphi$ is

M_\varphi=\{x\in M:\sigma_t^\varphi(x)=x\text{ for every }t\in\mathbb R\},

where $\sigma^\varphi$ is the modular automorphism group. It is therefore the fixed-point algebra of the modular flow. In particular, $M_\varphi$ is a von Neumann subalgebra of $M$ . The notation records the chosen weight: different weights on the same algebra can have different centralizers.

Tracial behavior

Elements of $M_\varphi$ commute with the weight in the following sense:

\varphi(xy)=\varphi(yx)

whenever $x\in M_\varphi$ and the products lie in the finite domain on which both sides are defined. Conversely, this commutation property characterizes the centralizer when formulated on the standard finite ideal of $\varphi$ . Thus the restriction of $\varphi$ to its centralizer is a faithful normal semifinite tracial weight Takesaki, vol. II, Chapter VIII, §2.

Examples and consequences

If $\varphi$ is a faithful normal semifinite trace, its modular group is trivial and $M_\varphi=M$ . For a faithful normal state on $M_n(\mathbb C)$ of the form $\varphi(x)=\operatorname{Tr}(\rho x)$ , with $\rho$ positive and invertible, the modular flow is $\sigma_t^\varphi(x)=\rho^{it}x\rho^{-it}$ . Hence

M_\varphi=\{x\in M_n(\mathbb C):x\rho=\rho x\}.

Repeated eigenvalues of $\rho$ produce matrix blocks in this centralizer; if $\rho$ is scalar, the centralizer is all of $M_n(\mathbb C)$ .

Conventions and scope

The centralizer is not the center of $M$ : its elements need only be fixed by the modular flow, and need not commute with every element of $M$ . For a nonfaithful normal weight, one commonly passes to its support corner before forming modular data. The faithfulness hypothesis in the core avoids this support-corner convention.

References

Masamichi Takesaki, Theory of Operator Algebras II, Springer, 2003. DOI record. Relevant: Chapter VIII, §2 on the centralizer of a weight.