Approximate identity in L1(G) - Imported Knowlpedia Content

Definition

Let $G$ be a locally compact group. An approximate identity in $L^1(G)$ is a net $(u_i)$ in the group convolution algebra $L^1(G)$ such that

\|u_i*f-f\|_1\longrightarrow 0 \quad\text{and}\quad \|f*u_i-f\|_1\longrightarrow 0

for every $f\in L^1(G)$ . It is bounded when $\sup_i\|u_i\|_1<\infty$ . Every locally compact group admits a bounded two-sided approximate identity with $u_i\geq 0$ , $\|u_i\|_1=1$ , and supports shrinking through the neighborhoods of the identity element.

Construction from small neighborhoods

Fix a left Haar measure on $G$ . For each identity neighborhood $U$ , one can choose $u_U\in C_c(G)$ satisfying

u_U\geq 0,\qquad \int_Gu_U(x)\,dx=1,\qquad \operatorname{supp}(u_U)\subseteq U.

Order the neighborhoods by reverse inclusion. Continuity of translations in $L^1(G)$ , followed by averaging against $u_U$ , gives the two convergence statements in the definition. The functions need not have a common formula: their normalization and concentration near the identity are the essential features.

Why it replaces an identity

Unless $G$ is discrete, the point mass at the identity is not an element of $L^1(G)$ , so convolution has no literal identity in the algebra. The net $(u_i)$ nevertheless recovers each $L^1$ -function in norm. It also detects nondegeneracy: for a bounded representation $T$ of $L^1(G)$ , the closed linear span of $T(L^1(G))H$ is the subspace on which $T(u_i)$ converges strongly to the identity.

Distinctions and conventions

An approximate identity is not a sequence in general; the directed set of neighborhoods may be uncountable. Nor is it a Dirac measure: each $u_i$ belongs to $L^1(G)$ , while the limiting point mass ordinarily belongs only to the measure algebra. Some authors require boundedness in the term “approximate identity”; the explicit norm-one construction removes that convention-dependent ambiguity here.

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. Publisher record. Relevant: Chapter 2 on Haar integration and approximate identities in $L^1(G)$ .
Hans Reiter and Jan D. Stegeman, Classical Harmonic Analysis and Locally Compact Groups, 2nd ed., Oxford University Press, 2000. DOI record. Relevant: the foundational theory of group convolution algebras.