Isometric Jordan Isomorphisms of Group Algebras

Let G and H be locally compact groups. We will show that each contractive Jordan isomorphism \(\Phi :L^1(G)\rightarrow L^1(H)\) $\mathrm{\Phi }:L^1(G)\to L^1(H)$ is either an isometric isomorphism or an isometric anti-isomorphism. We will apply this result to study isometric two-sided zero product preservers on group algebras and, further, to study local and approximately local isometric automorphisms of group algebras.