This paper investigates certain properties of n-Jordan homomorphisms between algebras (rings). We first show that every n-Jordan homomorphism T from a ring A into a commutative reduced algebra B is an n-homomorphism. For the case where A is a Banach algebra and B is a semisimple commutative Banach algebra, we prove that any linear n-Jordan homomorphism \(T : A \rightarrow B\) is automatically continuous, even if the algebras A and B are not unital. Furthermore, we show that every surjective, \(*\) -preserving linear n-Jordan homomorphism from a unital C*-algebra into a C*-algebra is norm-decreasing and hence automatically continuous. We then provide a complete characterization of bijective \(*\) -preserving linear n-Jordan homomorphisms between unital C*-algebras. Finally, we strengthen this result by proving that every continuous bijective linear \(*\) -preserving n-Jordan homomorphism between nonunital C*-algebras is an isometry. To achieve this, we show that the second adjoint of a continuous linear n-Jordan homomorphism between Arens regular Banach algebras is also an n-Jordan homomorphism.
