Tomita-Takesaki theory

M. Tomita [a4] defined the notion of a left Hilbert algebra as follows: An involutive algebra over the field of complex numbers, with involution , that admits an inner product satisfying the following conditions:

i) the mapping is continuous for every ;

ii) for all ;

iii) is total in the Hilbert space obtained by completion of .

iv) is a closeable conjugate-linear operator in . Let be a left Hilbert algebra in a Hilbert space . For any , let denote the unique continuous linear operator on such that , . The von Neumann algebra generated by is called the left von Neumann algebra of . Let be the closure of the mapping and let be the polar decomposition of . Then is an isometric involution and is a non-singular positive self-adjoint operator in satisfying and ; and are called the modular operator and the modular conjugation operator of , respectively. Let denote the set of vectors such that the mapping is continuous. For any , denote by the unique continuous extension of to . Let be the set of vectors such that the mapping is continuous. For any , denote by the unique continuous extension of to . Then is a left Hilbert algebra in , equipped with the multiplication and the involution , and is equivalently contained in , that is, and they have the same modular (conjugation) operators. The set is a left Hilbert algebra which is equivalently contained in and is a complex one-parameter group of automorphisms of , called the modular automorphism group. It satisfies the conditions:

a) , , ;

b) , , ;

c) , ;

d) , , is an analytic function on . 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 and for all . 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 with the operators , 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