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:
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—Milan Kundera (b. 1929)
“Unlike Boswell, whose Journals record a long and unrewarded search for a self, Johnson possessed a formidable one. His life in London—he arrived twenty-five years earlier than Boswell—turned out to be a long defense of the values of Augustan humanism against the pressures of other possibilities. In contrast to Boswell, Johnson possesses an identity not because he has gone in search of one, but because of his allegiance to a set of assumptions that he regards as objectively true.”
—Jeffrey Hart (b. 1930)