Let $G$ be a locally compact group (not only compact group) with the left Haar measure $\lambda$. Consider the convolution Banach algebra $L^1(G,\lambda)$. For which $f\in L^1(G,\lambda)$ the subspace $\{f*g*f: g\in L^1(G,\lambda)\}$ is not finite dimensional? For which one it is finite dimensional?
Convolution $*$: $$\psi *\phi(x)=\int_G\psi(y)\phi(y^{-1}x)d\lambda(y)\qquad (x\in G, \psi,\phi\in L^1(G,\lambda))$$