Is $L^{1}(\Omega,\mu)$ only an algebra when $\Omega$ is a group?

Question

Let $G$ be a locally compact group. Then we may snap our fingers, mention some measure theory results, and the group algebra $L^{1}(G)$ instantly appears.

Is there an example of the more generalized classical $L^{1}(\Omega, \mu)$ for a locally compact Hausdorff space $\Omega$, which turns out to be a Banach algebra such that either

1) $\Omega$ fails to be a locally compact group

or

2) $\mu$ is not a left Haar measure on $\Omega$?

Answer 1

Yes there is!

It is easy to construct absurd examples, but I guess you are looking for more interesting examples, such an example is provided by Zonal spherical functions where the translation operator is induced from a group and the measure is induced from a Haar measure.

Also, if I remember correctly M.A. Naimark (see his book Normed Rings) investigated abstract normed $L^1$-algebras based on an abstract translation operator.