Tomita-Takesaki theory is a collection of results in functional analysis concerning the theory of modular Hilbert algebras and, in particular, to a key result which details the construction of modular automorphisms on von Neumann algebras. Tomita first presented these results in two unpublished papers in 1967, as documented in the expanded exposition by Takesaki (1970). The general construction is as follows.
Given a von Neumann algebra 
on a Hilbert space 
with a vector 
which is both cyclic
and separating for 
, one can define an operator 
on 
by letting 
for all 
. Here, 
denotes the adjoint operator
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 the modular conjugation 
and the positive self-adjoint
modular operator 
associated to 
. In addition, 
is an antiunitary involution
and 
is a unitary 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.
For a normalized cyclic, separating vector 
, the state 
is faithful and normal and satisfies
the KMS condition with respect to the modular automorphism group associated
with 
(Summers 2005). This state depends on the chosen vector,
rather than being an intrinsic choice for every von
Neumann algebra.
