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.

This implies that there exists $(C(K)-C(K)^H)$ modules $M$ and $N$ such that $$ N^{op} \otimes_{C(K)} M \cong C(K)^H, \qquad M \otimes_{C(K)^H} N^{op} \cong C(K).$$

What are candidates for $N^{op}$ and $M$?

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.

Let $K$ be a compact group with a closed subgroup $H$. Let $C(K)$ be th 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.

This implies that there exists $(C(K)-C(K)^H)$ modules $M$ and $N$ such that $$ N^{op} \otimes_{C(K)} M \cong C(K)^H, \qquad M \otimes_{C(K)^H} N^{op} \cong C(K).$$

What are candidates for $N^{op}$ and $M$?

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.

This implies that there exists $(C(K)-C(K)^H)$ modules $M$ and $N$ such that $$ N^{op} \otimes_{C(K)} M \cong C(K)^H, \qquad M \otimes_{C(K)^H} N^{op} \cong C(K).$$

What are candidates for $N^{op}$ and $M$?