Approximate identity

Last updated January 31, 2023

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.

Definition

A right approximate identity in a Banach algebra A is a net $\{e_{\lambda }:\lambda \in \Lambda \}$ such that for every element a of A, $\lim _{\lambda \in \Lambda }\lVert ae_{\lambda }-a\rVert =0.$ Similarly, a left approximate identity in a Banach algebra A is a net $\{e_{\lambda }:\lambda \in \Lambda \}$ such that for every element a of A, $\lim _{\lambda \in \Lambda }\lVert e_{\lambda }a-a\rVert =0.$ An approximate identity is a net which is both a right approximate identity and a left approximate identity.

C*-algebras

For C*-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.

Convolution algebras

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.

Rings

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

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This is a glossary for the terminology in a mathematical field of functional analysis.

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v t e Spectral theory and ^*-algebras
Basic concepts	Involution/-algebra Banach algebra B-algebra C-algebra Noncommutative topology Projection-valued measure Spectrum Spectrum of a C-algebra Spectral radius Operator space
Main results	Gelfand–Mazur theorem Gelfand–Naimark theorem Gelfand representation Polar decomposition Singular value decomposition Spectral theorem Spectral theory of normal C*-algebras
Special Elements/Operators	Isospectral Normal operator Hermitian/Self-adjoint operator Unitary operator Unit
Spectrum	Krein–Rutman theorem Normal eigenvalue Spectrum of a C*-algebra Spectral radius Spectral asymmetry Spectral gap
Decomposition of a spectrum	(Continuous Point Residual) Approximate point Compression Discrete Spectral abscissa
Spectral Theorem	Borel functional calculus Min-max theorem Positive operator-valued measure Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
Special algebras	Amenable Banach algebra With an Approximate identity Banach function algebra Disk algebra Nuclear C*-algebra Uniform algebra Von Neumann algebra Tomita–Takesaki theory
Finite-Dimensional	Alon–Boppana bound Bauer–Fike theorem Numerical range Schur–Horn theorem
Generalizations	Dirac spectrum Essential spectrum Pseudospectrum Structure space (Shilov boundary)
Miscellaneous	Abstract index group Banach algebra cohomology Cohen–Hewitt factorization theorem Extensions of symmetric operators Limiting absorption principle Schröder–Bernstein theorems for operator algebras Sherman–Takeda theorem Unbounded operator
Examples	Wiener algebra
Applications	Almost Mathieu operator Corona theorem Hearing the shape of a drum (Dirichlet eigenvalue) Heat kernel Kuznetsov trace formula Lax pair Proto-value function Ramanujan graph Rayleigh–Faber–Krahn inequality Spectral geometry Spectral method Spectral theory of ordinary differential equations Sturm–Liouville theory Superstrong approximation Transfer operator Transform theory Weyl law Wiener–Khinchin theorem

v t e Banach space topics
Types of Banach spaces	Asplund Banach list Banach lattice Grothendieck Hilbert Inner product space Polarization identity (Polynomially) Reflexive Riesz L-semi-inner product (B Strictly Uniformly) convex Uniformly smooth (Injective Projective) Tensor product (of Hilbert spaces)
Banach spaces are:	Barrelled Complete F-space Fréchet tame Locally convex Seminorms/Minkowski functionals Mackey Metrizable Normed norm Quasinormed Stereotype
Function space Topologies	Dual Dual space Dual norm Operator Ultraweak Weak polar operator Strong polar operator Ultrastrong Uniform convergence
Linear operators	Adjoint Bilinear form operator sesquilinear (Un)Bounded Closed Compact on Hilbert spaces (Dis)Continuous Densely defined Fredholm kernel operator Hilbert–Schmidt Functionals positive Pseudo-monotone Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary
Operator theory	Banach algebras C-algebras Operator space Spectrum C-algebra radius Spectral theory of ODEs Spectral theorem Polar decomposition Singular value decomposition
Theorems	Anderson–Kadec Banach–Alaoglu Banach–Mazur Banach–Saks Banach–Schauder (open mapping) Banach–Steinhaus (Uniform boundedness) Bessel's inequality Cauchy–Schwarz inequality Closed graph Closed range Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Gelfand–Naimark Goldstine Hahn–Banach hyperplane separation Kakutani fixed-point Krein–Milman Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Parseval's identity Riesz's lemma Riesz representation Robinson-Ursescu Schauder fixed-point
Analysis	Abstract Wiener space Banach manifold bundle Bochner space Convex series Differentiation in Fréchet spaces Derivatives Fréchet Gateaux functional holomorphic quasi Integrals Bochner Dunford Gelfand–Pettis regulated Paley–Wiener weak Functional calculus Borel continuous holomorphic Measures Lebesgue Projection-valued Vector Weakly / Strongly measurable function
Types of sets	Absolutely convex Absorbing Affine Balanced/Circled Bounded Convex Convex cone (subset) Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx)) Linear cone (subset) Radial Radially convex/Star-shaped Symmetric Zonotope
Subsets /set operations	Affine hull (Relative) Algebraic interior (core) Bounding points Convex hull Extreme point Interior Linear span Minkowski addition Polar (Quasi) Relative interior
Examples	Absolute continuity AC $ba(\Sigma )$ c space Banach coordinate BK Besov $B_{p,q}^{s}(\mathbb {R} )$ Birnbaum–Orlicz Bounded variation BV Bs space Continuous C(K) with K compact Hausdorff Hardy H^p Hilbert H Morrey–Campanato $L^{\lambda ,p}(\Omega )$ ℓ^p $\ell ^{\infty }$ L^p $L^{\infty }$ weighted Schwartz $S\left(\mathbb {R} ^{n}\right)$ Segal–Bargmann F Sequence space Sobolev W^k,p Sobolev inequality Triebel–Lizorkin Wiener amalgam $W(X,L^{p})$
Applications	Differential operator Finite element method Mathematical formulation of quantum mechanics Ordinary Differential Equations (ODEs) Validated numerics