There are at least two distinct (though related) notions of the term Hilbert algebra in functional analysis.
In some literature, a linear manifold 
of a (not necessarily separable)
Hilbert space 
is a Hilbert algebra if the following
conditions are satisfied:
1. 
is dense in 
.
2. 
is a ring so that, for any 
, there is defined an element 
such that 
, 
, 
, and 
for any complex
number 
.
3. For any 
,
there exists an adjoint element 
such that 
, and 
.
4. For any 
,
there exists a positive number 
such that 
for all 
.
5. For every 
,
there exists a unique bounded linear
operator 
on 
such that 
for all 
.
Moreover, if 
for an element 
and for all 
,
then 
.
At least one author defines a Hilbert algebra to be a quasi-Hilbert algebra
 |
for which 
for all 
(Dixmier 1981).
