Measure algebra of a locally compact group

Let $G$ be a locally compact Hausdorff group. Its measure algebra $M(G)$ is the vector space of bounded regular complex Borel measures on $G$ , equipped with the total-variation norm and convolution determined by

\int_G f(z)\,d(\mu*\nu)(z) =\int_G\int_G f(xy)\,d\mu(x)\,d\nu(y)

for every $f\in C_0(G)$ . Equivalently, $(\mu*\nu)(E)$ is obtained by pushing the product measure forward along group multiplication. With this product, $M(G)$ is a unital Banach algebra whose identity is the point mass $\delta_e$ .

Point masses and the group law

For $x,y\in G$ , convolution satisfies

\delta_x*\delta_y=\delta_{xy}.

Thus $x\mapsto\delta_x$ embeds the group law into the invertible elements of $M(G)$ . The involution is characterized by $\delta_x^*=\delta_{x^{-1}}$ and, on test functions, by conjugation after inversion. Unlike the density formula for $L^1(G)$ , this measure-level description requires no chosen Haar measure.

Relation to the group algebra

After fixing a left Haar measure $m$ , each $f\in L^1(G,m)$ determines the measure $f\,dm$ . This identifies the $L^1$ group algebra isometrically with a closed two-sided ideal in $M(G)$ . Point masses at nondiscrete points are singular with respect to Haar measure, so $M(G)$ is generally strictly larger than $L^1(G)$ .

Examples and scope

If $G$ is discrete, every bounded measure is an absolutely summable family and $M(G)\cong\ell^1(G)$ . For nondiscrete $G$ , $M(G)$ simultaneously contains integrable densities and atomic measures. On an abelian group it is commutative; on a nonabelian group the point-mass calculation shows immediately that it need not be commutative.

References

E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Volume I, Springer, 1963. DOI record. Relevant: measure algebras and convolution of measures.
H. Reiter and J. D. Stegeman, Classical Harmonic Analysis and Locally Compact Groups, 2nd ed., Oxford University Press, 2000. Publisher record. Relevant: convolution algebras on locally compact groups.