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. For functional analysis as used in psychology see the Functional analysis (psychology article In Mathematics, especially Functional analysis, a Banach algebra, named after Stefan Banach, is an Associative algebra A over the This article is about nets in Topological spaces and not about ε-nets in Metric spaces In Topology and related areas of Mathematics In Mathematics, a sequence is an ordered list of objects (or events

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.

For C*-algebras, a right (or left) approximate identity is the same as an approximate identity. C*-algebras (pronounced "C-star" are an important area of research in Functional analysis, a branch of Mathematics. Every C*-algebra 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. In Mathematics, especially Functional analysis, a Hermitian element A of a C*-algebra is a positive element if its spectrum This is called the canonical approximate identity of a C*-algebra. Approximate identities of C*-algebras 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.

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). In Mathematics and in particular Functional analysis, convolution is a mathematical operation on two functions f and The Dirac delta or Dirac's delta is a mathematical construct introduced by the British theoretical physicist Paul Dirac. For example the Fejér kernels of Fourier series theory give rise to an approximate identity. In Mathematics, the Fejér kernel is used to express the effect of Cesàro summation on Fourier series. In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions

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