A modular Hilbert algebra is an involutive algebra 
over the field 
of complex
numbers with involution 
, an inner product 
and a one-parameter
group 
of automorphisms
on 
,
,
satisfying:
1. 
.
2. For all 
,
is bounded (hence, continuous)
on 
.
3. The vector space 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 left Hilbert algebras in that every modular
Hilbert algebra is a left 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 (Takesaki
1970).
