Editing Approximate identity

{{About|the Banach algebra concept||Approximation to the identity (disambiguation){{!}}Approximation to the identity}}
In [[mathematics]], particularly in [[functional analysis]] and [[ring theory]], an approximate identity is a [[net (mathematics)|net]] in a [[Banach algebra]] or [[ring (mathematics)|ring]] (generally without an identity) that acts as a substitute for an [[identity element]].

==Definition==

A '''right approximate identity''' in a Banach algebra ''A'' is a net <math>\{e_\lambda : \lambda \in \Lambda\}</math> such that for every element ''a'' of ''A'', <math>\lim_{\lambda\in\Lambda}\lVert ae_\lambda - a \rVert = 0.</math> Similarly, a '''left approximate identity''' in a Banach algebra ''A'' is a net <math>\{e_\lambda : \lambda \in \Lambda\}</math> such that for every element ''a'' of ''A'', <math>\lim_{\lambda\in\Lambda}\lVert e_\lambda a - a \rVert = 0.</math> An '''approximate identity''' is a net which is both a right approximate identity and a left approximate identity.

==C*-algebras==

For [[C*-algebra]]s, a right (or left) approximate identity consisting of [[self-adjoint]] elements is the same as an approximate identity. The net of all positive elements in ''A'' of norm ≤&thinsp;1 with its natural order is an approximate identity for any C*-algebra. This is called the '''canonical approximate identity''' of a C*-algebra. Approximate identities are not unique. For example, for [[compact operator]]s acting on a [[Hilbert space]], the net consisting of finite rank projections would be another approximate identity.

If an approximate identity is a [[sequence (mathematics)|sequence]], we call it a '''sequential approximate identity''' and a C*-algebra with a sequential approximate identity is called '''&sigma;-unital'''. Every [[separable space|separable]] C*-algebra is &sigma;-unital, though the [[converse (logic)|converse]] is false. A commutative C*-algebra is &sigma;-unital [[if and only if]] its [[spectrum of a C*-algebra|spectrum]] is [[&sigma;-compact]]. In general, a C*-algebra ''A'' is &sigma;-unital if and only if ''A'' contains a strictly positive element, i.e. there exists ''h'' in ''A''<sub>+</sub> such that the [[hereditary C*-subalgebra]] generated by ''h'' is ''A''.

One sometimes considers approximate identities consisting of specific types of elements. For example, a C*-algebra has [[Real rank (C*-algebras)#Real rank zero|real rank zero]] if and only if every hereditary C*-subalgebra has an approximate identity consisting of projections. This was known as property (HP) in earlier literature.

==Convolution algebras==

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 kernel]]s of [[Fourier series]] theory give rise to an approximate identity.

==Rings==

In ring theory, an approximate identity is defined in a similar way, except that the ring is given the [[discrete topology]] so that ''a'' = ''ae''<sub>λ</sub> for some λ.

A [[module (mathematics)|module]] over a ring with approximate identity is called '''non-degenerate''' if for every ''m'' in the module there is some λ with ''m'' = ''me''<sub>λ</sub>.

==See also==

* [[Mollifier]]
* [[Nascent delta function]]
* [[Summability kernel]]

{{Spectral theory}}
{{Functional analysis}}
{{Banach spaces}}

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[[Category:Banach algebras]]