A Hua-type theorem for Cayley algebras

We introduce the notion of \(*\) $\ast$ -Jordan homomorphism map to obtain a Hua type theorem on alternative \(*\) $\ast$ -algebras endowed with an involution, relating in which conditions a \(*\) $\ast$ -Jordan homomorphism preserves \(*\) $\ast$ -generalized inverses. Even more, we prove that the bijective unital continuous linear maps on octonion algebra \(\mathbb {O}\) $\mathbb{O}$ whose preserves \(*\) $\ast$ -generalized invertibility are \(*\) $\ast$ -Jordan homomorphism maps.