Let $K$ be a compact or a finite group with a closed subgroup $H$. Let $C(K)$ be the convolution algebra of continuous functions on $K$.


The Peter Weyl theorem asserts that the $\*$ algebra $C(K)$ and the subalgebra $C(K)^H$ of functions with
$$ f(h^{-1}k h) = f(k) \qquad h\in H$$
are Morita equivalent (wrong!). The algebra $C(K)^H$ can have additional modules.



Morita equivalence for compact groups
I don't believe this statement. In the finite case $C(K)$ is the usual group algebra. Let $K$ be nonabelian of order $6$ and let $H\subset K$ have order $2$. Then $C(K)^H$ is not Morita equivalent to $C(K)$: It is semisimple, but it has four simple modules up to isomorphism whereas $C(K)$ has three.


