View source for Tomita-Takesaki theory

M. Tomita [[#References|[a4]]] defined the notion of a left Hilbert algebra as follows: An involutive algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t1201501.png" /> over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t1201502.png" /> of complex numbers, with involution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t1201503.png" />, that admits an inner product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t1201504.png" /> satisfying the following conditions:

i) the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t1201505.png" /> is continuous for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t1201506.png" />;

ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t1201507.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t1201508.png" />;

iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t1201509.png" /> is total in the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015010.png" /> obtained by completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015011.png" />.

iv) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015012.png" /> is a closeable conjugate-linear operator in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015013.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015014.png" /> be a left Hilbert algebra in a [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015015.png" />. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015016.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015017.png" /> denote the unique continuous linear operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015018.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015020.png" />. The [[Von Neumann algebra|von Neumann algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015021.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015022.png" /> is called the left von Neumann algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015023.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015024.png" /> be the closure of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015025.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015026.png" /> be the polar decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015027.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015028.png" /> is an isometric involution and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015029.png" /> is a non-singular positive [[Self-adjoint operator|self-adjoint operator]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015030.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015032.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015034.png" /> are called the modular operator and the modular conjugation operator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015035.png" />, respectively. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015036.png" /> denote the set of vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015037.png" /> such that the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015038.png" /> is continuous. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015039.png" />, denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015040.png" /> the unique continuous extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015041.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015042.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015043.png" /> be the set of vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015044.png" /> such that the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015045.png" /> is continuous. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015046.png" />, denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015047.png" /> the unique continuous extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015048.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015049.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015050.png" /> is a left Hilbert algebra in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015051.png" />, equipped with the multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015052.png" /> and the involution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015053.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015054.png" /> is equivalently contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015055.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015056.png" /> and they have the same modular (conjugation) operators. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015057.png" /> is a left Hilbert algebra which is equivalently contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015059.png" /> is a complex one-parameter group of automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015060.png" />, called the modular automorphism group. It satisfies the conditions:

a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015063.png" />;

b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015066.png" />;

c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015068.png" />;

d) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015070.png" />, is an analytic function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015071.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015073.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015074.png" />. This theorem is called the Tomita fundamental theorem. M. Takesaki [[#References|[a3]]] arranged and deepened this theory and connected this theory with the Haag–Hugenholtz–Winnink theory [[#References|[a2]]] for equilibrium states for quantum statistical mechanics. After that, Tomita–Takesaki theory was developed by A. Connes [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015075.png" /> with the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120150/t12015076.png" />, A. van Daele [[#References|[a5]]] has simplified a discussion in the complicated Tomita–Takesaki theory.

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