Modular conjugation - Knowlpedia

Let $M\subseteq B(H)$ be a von Neumann algebra with cyclic separating vector $\Omega$ , and let $S$ be the associated Tomita operator. In the polar decomposition

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

the modular conjugation is the antiunitary operator $J:H\to H$ . It is conjugate-linear, satisfies $\langle J\xi,J\eta\rangle=\langle\eta,\xi\rangle$ , and is an involution: $J^2=1$ . Moreover $J\Omega=\Omega$ . Both $J$ and the positive modular operator $\Delta$ are determined by the pair $(M,\Omega)$ ; the notation $J$ does not denote a choice of complex structure.

Relation to the commutant

The Tomita–Takesaki theorem identifies the conjugated algebra as

JMJ=M',

where $M'$ is the commutant of $M$ . Thus $x\mapsto JxJ$ is a conjugate-linear multiplicative bijection from $M$ onto $M'$ ; equivalently, $x\mapsto Jx^*J$ is a linear $*$ -anti-isomorphism. This conclusion is a theorem, not merely a consequence of antiunitarity.

Interaction with modular time

The polar factors satisfy

J\Delta J=\Delta^{-1} \quad\text{and}\quad J\Delta^{it}J=\Delta^{it}.

Consequently $J$ exchanges the algebra with its commutant while respecting the unitary modular evolution in the appropriate conjugate-linear sense. When the vector state is tracial, $\Delta=1$ , but $J$ can still be nontrivial: it implements passage from left to right multiplication in the standard representation.

Example and conventions

For $M=M_n(\mathbb C)$ acting on its Hilbert–Schmidt space with trace vector, $J(x)=x^*$ . Left multiplication by $a$ is carried to right multiplication by $a^*$ , exhibiting $JMJ=M'$ .

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter VI, §1 on the polar decomposition of the Tomita operator.