Let 
be an involutive algebra over the field 
of complex
numbers with involution 
. Then 
is a left Hilbert algebra if 
has an inner product 
satisfying:
1. For all 
,
is bounded on 
.
2. 
.
3. The involution 
is closable.
4. The linear span 
of products 
, 
, is a dense subalgebra
of 
.
Left Hilbert algebras are historically known as generalized Hilbert algebras (Takesaki 1970).
A basic result in functional analysis says that if the involution map 
on a left Hilbert algebra 
is an antilinear isometry
with respect to the inner product 
, then 
is also a right Hilbert
algebra with respect to the involution 
. The converse also holds.
