Chemistry - Approximate identity

In functional analysis, a right approximate identity in a Banach algebra A is a net (or a sequence)

$\{\,e_\lambda : \lambda \in \Lambda\,\}$

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.

Every C*-algebra $A$ 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.

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.

Categories: Functional analysis