Rigidity of Banach *-algebras

It is well known that two non-isomorphic groups (groupoids) can produce isomorphic C*-algebras. That is, group (groupoid) C*-algebras are not rigid. This is not the case of the L^p-operator algebras associated to locally compact groups ( effective groupoids) where the isomorphic class of the group (groupoid) uniquely determines up to isometric isomorphism the associated L^p-algebras. Thus, L^p-operator algebras are rigid.  Liao and Yu introduced a class of Banach *-algebras associated to locally compact groups. We will see that this family of Banach *-algebras are also rigid.