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.