I asked this initially in math.stackexchange, 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.