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:
“the dull thunder of approximate words.”
—Thom Gunn (b. 1929)
“The “female culture” has shifted more rapidly than the “male culture”; the image of the go-get ‘em woman has yet to be fully matched by the image of the let’s take-care-of-the-kids- together man. More important, over the last thirty years, men’s underlying feelings about taking responsibility at home have changed much less than women’s feelings have changed about forging some kind of identity at work.”
—Arlie Hochschild (20th century)