Modular operator - Imported Knowlpedia Content

Definition

Let $M\subseteq B(H)$ be a von Neumann algebra with cyclic separating vector $\Omega$ , and let $S$ be its Tomita operator. The modular operator of $(M,\Omega)$ is

\Delta=S^*S.

It is a positive self-adjoint operator with trivial kernel and generally is unbounded. Its positive square root occurs in the polar decomposition

S=J\Delta^{1/2},

where $J$ is the modular conjugation. The spectral calculus defines a strongly continuous unitary group $t\mapsto\Delta^{it}$ , even when $\Delta$ or $\Delta^{-1}$ is unbounded. The operator depends on the chosen pair $(M,\Omega)$ .

Modular dynamics

The Tomita–Takesaki theorem states that

\Delta^{it}M\Delta^{-it}=M,\qquad t\in\mathbb R.

Hence $\sigma_t^\Omega(x)=\Delta^{it}x\Delta^{-it}$ defines a one-parameter group of $*$ -automorphisms of $M$ . The invariance of $M$ is the deep modular theorem; it does not follow from spectral calculus alone Takesaki, Chapter VI, §1.

Spectral relations

Modular conjugation reverses the positive generator:

J\Delta J=\Delta^{-1}.

The vector $\Omega$ lies in the kernel of $\Delta-1$ , because $S\Omega=\Omega=S^*\Omega$ . If the vector functional $\langle\,\cdot\,\Omega,\Omega\rangle$ is tracial, then $\Delta=1$ . Conversely, in the cyclic separating setting, $\Delta=1$ implies that this vector functional is a trace.

Finite-dimensional model

Let $M=M_n(\mathbb C)$ act by left multiplication on Hilbert–Schmidt matrices, and represent a faithful state by an invertible density matrix $\rho$ . Under the standard identification, the modular operator acts as

\Delta(x)=\rho x\rho^{-1}.

Thus $\Delta^{it}(x)=\rho^{it}x\rho^{-it}$ . It equals the identity exactly when $\rho$ is scalar, illustrating how modular data measure the failure of a faithful state to be tracial.

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter VI, §1 on the modular operator and its unitary group.