Let 
be an involutive algebra over the field 
of complex
numbers with involution 
. Then 
is a modular Hilbert algebra if 
has an inner product 
and a one-parameter
group 
of automorphisms
on 
,
,
satisfying:
1. 
.
2. For all 
,
is bounded (hence, continuous)
on 
.
3. The linear span 
of products 
, 
, is a dense subalgebra
of 
.
4. 
for all 
,
.
5. 
.
6. 
.
7. 
is an entire function of 
on 
.
8. For every real number 
, the set 
is dense in 
.
The group 
is called the group of modular automorphisms.
Note that the definition of modular Hilbert algebras is closely related to that of generalized Hilbert algebras in that every
modular Hilbert algebra is a generalized Hilbert algebra provided that it satisfies
one additional condition, namely that the involution 
is closable
as a linear operator on the real pre-Hilbert
space 
.
This relationship is due, in part, to the fact that the properties of both structures
were at the core of Tomita's original exposition of what is today the heart of Tomita-Takesaki theory.
