The phrase Tomita-Takesaki theory refers to a specific collection of results proven within the field of functional analysis regarding the theory of modular Hilbert algebras and, in particular, to a key result which details the construction of modular automorphisms on von Neumann algebras. These results were first presented in two unpublished papers by Tomita from 1967 and were brought to the public eye in the expanded exposition by Takesaki in 1970. The general construction is as follows.
Given a von Neumann algebra 
on a Hilbert space 
which contains a vector 
which is both cyclic
and separating for 
, one can define an operator 
on 
by letting 
for all 
. Here, 
denotes the dual of
.
A straightforward computation shows that 
extends to a closed antilinear operator 
defined on a dense subset
of 
,
and by applying a so-called polar decomposition
to 
,
it follows that
 |
for unique operators 
(called the modular operator) and 
(called modular conjugation or modular involution) associated
to 
.
In addition, 
is self-dual so that 
and 
is a unitary operator
for each 
.
In the context of this framework, the main result lying at the foundation of Tomita-Takesaki theory says that for any von Neumann algebra 
with a cyclic, separating vector 
, 
, 
for all 
, and 
where 
is the collection of all bounded,
linear operators on 
which commute with all elements
of 
.
Moreover, defining an operator 
on elements 
by 
and taking the closure 
of 
,
one also has that 
, 
, and 
.
What today exists as the modular theory of Tomita-Takesaki begins with the above result, as well as a number of important notions stemming from Tomita's original
presentation thereof. From this result, one can consider the one-parameter collection
of automorphisms 
of 
induced by the unitaries 
, 
, by defining
 |
for all 
and all 
.
This collection forms a group called the modular automorphism group on 
relative to 
, and this group plays a significant role in a variety
of fields including functional analysis and mathematical physics.
One example of the interplay between this modular automorphism group and modern physics comes from the fact that every von Neumann algebra 
has a naturally induced state 
which satisfies a number of desired properties with respect
to the above-defined modular automorphism group, not the least significant of which
is the so-called Kubo-Martin-Schwinger boundary condition.
As a result, the modular automorphism group inherits a number of significant properties
from the area of quantum statistical mechanics and other related areas.
