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Approximate Identity

An approximate identity for a normed algebra A is a net {e_i} in A such that, for every a in A, e_ia->a and ae_i->a (Foralewski and Piszczek 2023, p. 13). If, in addition,

sup_(i)||e_i||<infty,

then {e_i} is called a bounded approximate identity (Conway 1990, p. 195). Though this definition makes sense for any normed algebra A, it is usually used when A is either a Banach algebra or a local Banach algebra.

If only e_ia->a (ae_i->a, respectively), then {e_i} is called a left (respectively, right) approximate identity.

See also

Algebra, Banach Algebra, Local Banach Algebra, Net, Normed Space, Real Normed Algebra

Portions of this entry contributed by Christopher Stover

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References

Blackadar, B. K-Theory for Operator Algebras, 2nd ed. New York: Cambridge University Press, 1998.Conway, J. B. A Course in Functional Analysis, 2nd ed. New York: Springer-Verlag, 1990.Foralewski, P. and Piszczek, K. "On Orlicz Sequence Algebras." J. Inequal. Appl. 2023, Article 84, 2023. https://doi.org/10.1186/s13660-023-02978-2.

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Approximate Identity

Cite this as:

Weisstein, Eric W., with contributions by Christopher Stover. "Approximate Identity." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ApproximateIdentity.html

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