A right approximate identity in a Banach algebra A is a net
Similarly, a left approximate identity in a Banach algebra A is a net
The net of all positive elements in A of norm ≤ 1 with its natural order is an approximate identity for any C*-algebra.
For example, for compact operators acting on a Hilbert space, the net consisting of finite rank projections would be another approximate identity.
In general, a C*-algebra A is σ-unital if and only if A contains a strictly positive element, i.e. there exists h in A+ such that the hereditary C*-subalgebra generated by h is A.
One sometimes considers approximate identities consisting of specific types of elements.
For example, a C*-algebra has real rank zero if and only if every hereditary C*-subalgebra has an approximate identity consisting of projections.
An approximate identity in a convolution algebra plays the same role as a sequence of function approximations to the Dirac delta function (which is the identity element for convolution).
For example, the Fejér kernels of Fourier series theory give rise to an approximate identity.