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Concept# Approximate identity

Summary

In mathematics, particularly in functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring (generally without an identity) that acts as a substitute for an identity element.
A right approximate identity in a Banach algebra A is a net such that for every element a of A, Similarly, a left approximate identity in a Banach algebra A is a net such that for every element a of A, An approximate identity is a net which is both a right approximate identity and a left approximate identity.
For C*-algebras, 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 ≤ 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 operators acting on a Hilbert space, the net consisting of finite rank projections would be another approximate identity.
If an approximate identity is a sequence, we call it a sequential approximate identity and a C*-algebra with a sequential approximate identity is called σ-unital. Every separable C*-algebra is σ-unital, though the converse is false. A commutative C*-algebra is σ-unital if and only if its spectrum is σ-compact. In general, a C*-algebra A is σ-unital if and only if A contains a strictly positive element, i.e. there exists h in A+ 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 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.
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.

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Related concepts (2)

Approximate identity

In mathematics, particularly in functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring (generally without an identity) that acts as a substitute for an identity element. A right approximate identity in a Banach algebra A is a net such that for every element a of A, Similarly, a left approximate identity in a Banach algebra A is a net such that for every element a of A, An approximate identity is a net which is both a right approximate identity and a left approximate identity.

Dirac delta function

In mathematical physics, the Dirac delta distribution (δ distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. The current understanding of the unit impulse is as a linear functional that maps every continuous function (e.g., ) to its value at zero of its domain (), or as the weak limit of a sequence of bump functions (e.g.

Related lectures (1)

Approximation by Smooth Functions

Discusses approximation by smooth functions and the convergence of function sequences in normed vector spaces.