- Approximate identity
In

functional analysis , a**right approximate identity**in aBanach algebra , "A", is a net (or asequence ):$\{,e\_lambda\; :\; lambda\; in\; Lambda,\}$

such that for every element, "a", of "A", the net (or sequence)

:$\{,ae\_lambda:lambda\; in\; Lambda,\}$

has limit "a".

Similarly, a

**left approximate identity**is a net:$\{,e\_lambda\; :\; lambda\; in\; Lambda,\}$

such that for every element, "a", of "A", the net (or sequence)

:$\{,e\_lambda\; a:\; lambda\; in\; Lambda,\}$

has limit "a".

An

**approximate identity**is a right approximate identity which is also a left approximate identity.For

C*-algebra s, 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 theDirac delta function (which is the identity element for convolution). For example theFejér kernel s ofFourier series theory give rise to an approximate identity.

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