In mathematics, particularly in

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...

and

ring theory In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...

, an approximate identity is a net in a

Banach algebra In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...

or ring (generally without an identity) that acts as a substitute for an

identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures s ...

Definition

A right approximate identity in a Banach algebra ''A'' is a net

\

such that for every element ''a'' of ''A'',

\lim_\lVert ae_\lambda - a \rVert = 0.

Similarly, a left approximate identity in a Banach algebra ''A'' is a net

\

such that for every element ''a'' of ''A'',

\lim_\lVert e_\lambda a - a \rVert = 0.

An approximate identity is a net which is both a right approximate identity and a left approximate identity.

C*-algebras

For

C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continu ...

s, a right (or left) approximate identity consisting of

self-adjoint In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection ''C'' of elements of a sta ...

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 operator In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact ...

s acting on a

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...

, the net consisting of finite rank projections would be another approximate identity. If an approximate identity is a

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...

, 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 In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bi ...

its

spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of color ...

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.

Convolution algebras

An approximate identity in a

convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution' ...

algebra plays the same role as a sequence of function approximations to the

Dirac delta function In mathematics, the Dirac delta distribution ( 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 ...

(which is the identity element for convolution). For example, the

Fejér kernel In mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. ...

s of

Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...

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 In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest t ...

so that ''a'' = ''ae''_λ for some λ. A 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''_λ.

Definition

C*-algebras

Convolution algebras

Rings

See also