An approximate identity is a right approximate identity which is also a left approximate identity.
For C*-algebras, a right (or left) approximate identity is the same as an approximate identity. Every C*-algebra has an approximate identity of positive elements of norm ≤ 1; indeed, the net of all positive elements of norm ≤ 1; in A with its natural order always suffices. This is called the canonical approximate identity of a C*-algebra. Approximate identities of C*-algebras are not unique. For example, for compact operators acting on a Hilbert space, the net consisting of finite rank projections would be another approximate identity.
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.
Read more about Approximate Identity: Ring Theory
Famous quotes containing the words approximate and/or identity:
“A worker may be the hammer’s master, but the hammer still prevails. A tool knows exactly how it is meant to be handled, while the user of the tool can only have an approximate idea.”
—Milan Kundera (b. 1929)
“Whether outside work is done by choice or not, whether women seek their identity through work, whether women are searching for pleasure or survival through work, the integration of motherhood and the world of work is a source of ambivalence, struggle, and conflict for the great majority of women.”
—Sara Lawrence Lightfoot (20th century)