Tomita-Takesaki theory

M. Tomita [a4] defined the notion of a left Hilbert algebra as follows: An involutive algebra $\mathcal{A}$ over the field $\mathbf{C}$ of complex numbers, with involution $\xi \in \mathcal{A} \mapsto \xi ^ { \# } \in \mathcal{A}$, that admits an inner product $( \xi | \eta )$ satisfying the following conditions:

i) the mapping $\eta \in \mathcal{A} \mapsto \xi \eta \in \mathcal{A}$ is continuous for every $\xi \in \mathcal{A}$;

ii) $( \xi \eta _ { 1 } | \eta _ { 2 } ) = ( \eta _ { 1 } | \xi ^ { \# } \eta _ { 2 } )$ for all $\xi , \eta _ { 1 } , \eta _ { 2 } \in \mathcal{A}$;

iii) $\mathcal{A} ^ { 2 } \equiv \{ \xi \eta : \xi , \eta \in \mathcal{A} \}$ is total in the Hilbert space $\mathcal{H}$ obtained by completion of $\mathcal{A}$.

iv) $\xi \in \mathcal{A} \mapsto \xi ^ { \# } \in \mathcal{A}$ is a closeable conjugate-linear operator in $\mathcal{H}$. Let $\mathcal{A}$ be a left Hilbert algebra in a Hilbert space $\mathcal{H}$. For any $\xi \in \mathcal{A}$, let $\pi ( \xi )$ denote the unique continuous linear operator on $\mathcal{H}$ such that $\pi ( \xi ) \eta = \xi \eta$, $\eta \in \mathcal{A}$. The von Neumann algebra ${\cal L} ( A )$ generated by $\pi ( \mathcal{A} )$ is called the left von Neumann algebra of $\mathcal{A}$. Let $S$ be the closure of the mapping $\xi \in \mathcal{A} \rightarrow \xi ^ { \# } \in \mathcal{A}$ and let $S = J \Delta ^ { 1 / 2 }$ be the polar decomposition of $S$. Then $J$ is an isometric involution and $\Delta$ is a non-singular positive self-adjoint operator in $\mathcal{H}$ satisfying $S = J \Delta ^ { 1 / 2 } = \Delta ^ { - 1 / 2 } J$ and $S ^ { * } = J \Delta ^ { - 1 / 2 } = \Delta ^ { 1 / 2 } J$; $\Delta$ and $J$ are called the modular operator and the modular conjugation operator of $\mathcal{A}$, respectively. Let $\mathcal{A} ^ { \prime }$ denote the set of vectors $\eta \in \mathcal{D} ( S ^ { * } )$ such that the mapping $\xi \in \mathcal{A} \rightarrow \pi ( \xi ) \eta$ is continuous. For any $\eta \in \mathcal{A} ^ { \prime }$, denote by $\pi ^ { \prime } ( \eta )$ the unique continuous extension of $\xi \rightarrow \pi ( \xi ) \eta$ to $\mathcal{H}$. Let $\mathcal{A} ^ { \prime \prime }$ be the set of vectors $\xi \in \mathcal{D} ( S )$ such that the mapping $\eta \in \mathcal{A} ^ { \prime } \rightarrow \pi ^ { \prime } ( \eta ) \xi$ is continuous. For any $\xi \in \mathcal{A} ^ { \prime \prime }$, denote by $\pi ( \xi )$ the unique continuous extension of $\eta \rightarrow \pi ^ { \prime } ( \eta ) \xi$ to $\mathcal{H}$. Then $\mathcal{A} ^ { \prime \prime }$ is a left Hilbert algebra in $\mathcal{H}$, equipped with the multiplication $\xi _ { 1 } \xi _ { 2 } \equiv \pi ( \xi _ { 1 } ) \xi _ { 2 }$ and the involution $\xi \rightarrow \xi ^ { \# } \equiv S \xi$, and $\mathcal{A}$ is equivalently contained in $\mathcal{A} ^ { \prime \prime }$, that is, $\mathcal{A} \subset \mathcal{A} ^ { \prime \prime }$ and they have the same modular (conjugation) operators. The set $\mathcal{A} _ { 0 } \equiv \left\{ \xi \in A ^ { \prime \prime } : \xi \in \cap _ { \alpha \in \text{C} } \mathcal{D} ( \Delta ^ { \alpha } ) \right\}$ is a left Hilbert algebra which is equivalently contained in $\mathcal{A} ^ { \prime \prime }$ and $\{ \Delta ^ { \alpha } : \alpha \in \mathbf{C} \}$ is a complex one-parameter group of automorphisms of $\mathcal{A} _ { 0 }$, called the modular automorphism group. It satisfies the conditions:

a) $( \Delta ^ { \alpha } \xi ) ^ { \# } = \Delta ^ { - \overline { \alpha } } \xi ^ { \# }$, $\xi \in \mathcal{A} _ { 0 }$, $\alpha \in \mathbf{C}$;

b) $( \Delta ^ { \alpha } \xi | \eta ) = ( \xi | \Delta ^ { \overline { \alpha } } \eta )$, $\xi , \eta \in \mathcal{A} _ { 0 }$, $\alpha \in \mathbf{C}$;

c) $( \Delta \xi ^ { \# } | \eta ^ { \# } ) = ( \eta | \xi )$, $\xi , \eta \in \mathcal{A} _ { 0 }$;

d) $\alpha \in \mathbf{C} \rightarrow ( \Delta ^ { \alpha } \xi | \eta )$, $\xi , \eta \in \mathcal{A} _ { 0 }$, is an analytic function on $\mathbf{C}$. Such a left Hilbert algebra is called a modular Hilbert algebra (or Tomita algebra). Using the theory of modular Hilbert algebras, M. Tomita proved that $J \mathcal{L} ( \mathcal{A} ) J = \mathcal{L} ( \mathcal{A} ) ^ { \prime }$ and $\Delta ^ { i t } \mathcal{L} ( \mathcal{A} ) \Delta ^ { - i t } = \mathcal{L} ( \mathcal{A} )$ for all $t \in \mathbf{R}$. This theorem is called the Tomita fundamental theorem. M. Takesaki [a3] arranged and deepened this theory and connected this theory with the Haag–Hugenholtz–Winnink theory [a2] for equilibrium states for quantum statistical mechanics. After that, Tomita–Takesaki theory was developed by A. Connes [a1], H. Araki, U. Haagerup and the others, and has contributed to the advancement of the structure theory of von Neumann algebras, non-commutative integration theory, and quantum physics. Using an integral formula relating the resolvent of the modular operator $\Delta$ with the operators $\Delta ^ { i t }$, A. van Daele [a5] has simplified a discussion in the complicated Tomita–Takesaki theory.

References

[a1]	A. Connes, "Une classification des facteurs de type III" Ann. Sci. Ecole Norm. Sup. , 6 (1973) pp. 133–252
[a2]	R. Haag, N.M. Hugenholts, M. Winnink, "On the equilibrium states in quantum mechanics" Comm. Math. Phys. , 5 (1967) pp. 215–236
[a3]	M. Takesaki, "Tomita's theory of modular Hilbert algebras and its applications" , Lecture Notes Math. , 128 , Springer (1970)
[a4]	M. Tomita, "Standard forms of von Neumann algebras" , The Vth Functional Analysis Symposium of Math. Soc. Japan, Sendai (1967)
[a5]	A. Van Daele, "A new approach to the Tomita–Takesaki theory of generalized Hilbert algebras" J. Funct. Anal. , 15 (1974) pp. 378–393