Bounded approximate identities in one-sided ideals of $L_1(G)$

Question

Let $G$ be a locally compact group and consider its $L_1$-group algebra, that is, $L_1(G)$. It is easily seen that $L_1(G)$ has a two-sided bounded approximate identity: just use a basis of compact neigbourhoods of $e\in G$ to construct one.

My question is whether of the same is true for closed right/left ideals of $L_1(G)$?

• Does every closed one-sided ideal of $L_1(G)$ contain a bounded one-sided approximate identity?

I know that left/right they are right/left-translation invariant subspaces of $L_1(G)$ but it is not clear to me if it helps.

What if we assume that the group is amenable?

Answer 1

In general the answer is no. If $G$ is compact and abelian then $I$ is a left closed ideal with bounded approximate identity iff $I=L_1(G)*\mu$ for some idempotent $\mu\in M(G)$ iff $I$ is a kernel of coset ring of closed subsets in dual group of $G$. Taking a kernel of the set that doesn't belong to that coset ring you get (not much explicit) example of ideal without bounded approximate identity. For details see section 5.6 in A сourse in commutative Banach algebras. Eberhard Kaniuth.

Even amenability doesn't help since all compact group are ameanable.

Answer 2

When you suppose that $G$ is amenable we have $L^1(G)$ as an amenable Banach algebra. Now if any ideal of it has left or right bounded approximate identity then it has to be weakly complemented in $A$ and vice versa. Also if ideal has two sided bounded approximate identity then it has to be amenable and vice versa.