Let $G$ be a finite group, and consider the composition $X \* Y$ on $\mathbb{C}G$ defined by $$(\sum\_g \alpha\_g u\_g) \* (\sum\_g \beta\_g u\_g) = \sum\_g \alpha\_g \beta\_g u\_g.$$ This composition can be reformulated as a *convolution* using the Fourier transform.


*Question*: Let $X,Y \in \mathbb{C}G$ be positive elements. Is $X \* Y$ also positive? 


*Remark*: It is an application of Theorem 4.1 in <http://dx.doi.org/10.1090/tran/6582> to finite groups theory, but we are here interested in a purely group-theoretic reference or proof.


It is like to [Schur product Theorem](https://en.wikipedia.org/wiki/Schur_product_theorem) because $X \* Y$ is an entrywise product like the [Schur product](https://en.wikipedia.org/wiki/Hadamard_product_(matrices)).


*Bonus question*: Can we extend to any locally compact group?



A Schur-like product theorem on groups
The answer is positive. For ease of notation, let me write $a=\sum\_g \alpha\_g u\_g$ and $b = \sum\_g \beta\_g u\_g$.


Let's equip $\mathbb{C}G$ with the positive inner product induced by the standard basis. In terms of the usual normalized trace on a group algebra, this inner product is given by $(x,y)\mapsto\mathrm{tr}(x^\*y)$. So when you decompose $\mathbb{C}G$ into a direct sum of matrix algebras, this is just the usual Hilbert-Schmidt inner product. In particular, the cone of positive elements is nicely self-dual: we have $y\geq 0$ if and only if $\mathrm{tr}(xy)\geq 0$ for all $x\geq 0$. Hence it is enough to show that $\mathrm{tr}\left(x(a \ast b)\right)\geq 0$ for all $x\geq 0$.


Now your composition operation $\*:\mathbb{C}G\otimes\mathbb{C}G\to\mathbb{C}G$ is the adjoint of the usual comultiplication $\Delta:\mathbb{C}G\to\mathbb{C}G\otimes\mathbb{C}G$, meaning that
$$
\mathrm{tr}\left(x(a \ast b)\right) = \mathrm{tr}(\Delta(x)(a\otimes b)).
$$
Finally, we have $\Delta(x)\geq 0$ since $\Delta$ is a $\*$-homomorphism, and $a\otimes b\geq 0$ since $a\geq 0$ and $b\geq 0$ by assumption.


I don't know about the generalization to the locally compact case.


