Identity and approximate identity in the L^1 group algebra

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Problem VII.2.11 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section VII.2 "Ideals and Quotients"

If $G$ is a locally compact group, show that $L^1(G)$ (1.11) has an approximate identity.

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Problem VII.9.1 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section VII.9 "The Group Algebra of a Locally Compact Abelian Group"

Prove that if $L^1(G)$ has an identity, then $G$ is discrete.

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References

[Con07] Conway J.B. A Course in Functional Analysis. 2nd ed., Springer, 2007.