Approximate identity

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This article is about the Banach algebra concept. For other uses, see Approximation to the identity.

In functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring (possibly without an identity) that acts as a substitute for an identity element.

More precisely, 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.

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. 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. 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). For example the Fejér kernels of Fourier series theory give rise to an approximate identity.

Ring theory[edit]

In ring theory an approximate identity is defined in a similar way, except that the ring is given the discrete topology so that a=aeλ 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=meλ.

See also[edit]