Let us consider the measure algebra $M(\mathbb{R})$ consisting of all Radon measures on reals.
Let $\mu$ be a Radon measure in $M(\mathbb{R})$ and $\delta\_0$ be the point mass measure concentrated on 0, which is also the multiplicative identity of $M(\mathbb{R})$.



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> Q. Does there exist any strictly positive real number $r>0$ and Radon measures $\mu\_1,\cdots \mu\_n$ satisfying the following idenity? $$\mu^\*\*\mu=r\delta\_0+\sum\_1^n\mu\_i^\*\*\mu\_i.$$
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Remark. $M(\mathbb{R})=C\_0(\mathbb{R})^\*$ is a unital dual Banach $\*$-algebra. The involution on $M(\mathbb{R})$ is defined as follow
$$\langle \mu^\*,f\rangle=\int f(t)\overline{d\mu(-t)}.$$



An equation in the convolution measure algebra on reals
The answer is no in general. Indeed, let $\hat\mu$ denote the Fourier transform of $\mu$, so that $\hat\mu(t)=\int e^{itx}\mu(dx)$ for real $t$. Then the equality
$$\mu^\*\*\mu=r\delta\_0+\sum\_1^n\mu\_i^\*\*\mu\_i$$
would imply
$$|\hat\mu|^2=r+\sum\_1^n|\hat\mu\_i|^2\ge r>0.$$
Taking now any $\mu$ with $\hat\mu(t)\to0$ as $t\to\infty$ gives a contradiction.


