I asked this initially in [math.stackexchange][1], but it disappeared almost immediately, so I hope it will be proper to aks this here.

Hewitt and Ross define *trigonometric polynomial* on a locally compact group $G$ as a (finite) linear combination of matrix elements of continuous unitary irreducible representations $\pi_i:G\to B(H_i)$ (not necessarily finite dimensional) of $G$:
$$
f(t)=\sum_{i=1}^n \lambda_i\cdot\langle\pi_i(t)x_i,y_i\rangle,\qquad t\in G, \quad \lambda_i\in{\mathbb C},\quad x_i,y_i\in H_i.
$$
If $G$ is abelian or compact then the space ${\tt Trig}(G)$ of trigonometric polynomials on $G$ is an algebra with respect to the pointwise multiplication. 

This is strange, I can't find mentionings of the same proposition in the non-abelian and non-compact case. Is it possible that in general case (for arbitrary locally compact group $G$) the space ${\tt Trig}(G)$ is not an algebra?

I would be grateful, if somebody could advice reading on this theme.

  [1]: https://math.stackexchange.com/questions/1118632/trigonometric-polynomials-on-non-compact-and-non-abelian-groups