Jordan Left $\alpha$-centralizers on Algebras with Applications to Group Algebras

We prove that every Jordan left $\alpha$-centralizer from an algebra $A$ with a right identity into an arbitrary algebra $B$ is a left $\alpha$-centralizer. This implies all Jordan homomorphisms between such algebras are homomorphisms. We extend this result to continuous Jordan left $\alpha$-centralizers when $A$ has a bounded left approximate identity. For the group algebra $L^1(G)$, we characterize weakly compact Jordan left $\alpha$-centralizers when $\alpha$ is continuous and surjective, showing $L^1(G)$ admits a weakly compact epimorphism if and only if $G$ is finite. Consequently, the existence of a non-zero $\alpha$-derivation on $L^1(G)$ is equivalent to $G$ being compact and non-abelian.