# Convergence on locally compact groups with an additional condition

This question concerns locally compact groups equipped with Haar measure, $$(G,\lambda)$$. For a class of such groups, there exists an approximate identity $$F_\nu$$ such that the map $$f\in L^1(G)\mapsto f\ast F_\nu\in L^1(G)$$ is of finite rank for each $$\nu$$. A well known example is given by the Fejér kernels on the torus. Also for a certain class of compact groups which include finite dimensional unitary groups and its closed subgroups, there exist also such approximate identities: see for example Theorem 44.25, Hewitt & Ross, Abstract Harmonic Analysis, 2nd ed.)

I'm interested in knowing which other locally compact groups admit such approximate identities satisfying this additional finite rankness property. that is:

For which locally compact groups there exists an approximate identity $$F_n$$ such that the map $$f\in L^1(G)\mapsto f\ast F_\nu\in L^1(G)$$ is of finite rank for each $$\nu$$?

A more specific subquestion is:

Are there such approximate identities for all compact abelian groups?

NOTE: there is an answer here about the presence of approximate identities in $$L^1(G)$$, but in that thread the matter of finite rankness is not addressed.