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.