Certain properties of operator algebras have been studied such as boundedness, positivity, surjectivity, linearity, invertibility, numerical range, numerical radius and idempotent property. Of great interest is the study of spectrum of linear mappings. It is therefore necessary to characterize Jordan homomorphisms on semisimple Banach algebras in terms of their spectrum. The objectives of the study are to: Investigate whether Jordan homomorphisms on semisimple Banach algebras are spectral isometries; Investigate whether Jordan homomorphisms are unital surjections on semisimple Banach algebras and to establish the relationship between unital surjections and spectral isometries on semsimple Banach algebras. For us to achieve our objectives we used Kadison's theorem, Gelfand theory and Nagasawa's theorem. The results obtained show that Jordan homomorphism is spectral isometry if it preserves nilpotency also is unital surjection if it preserves Jordan zero products and finally is unital surjective spectral isometry if it preserves commutativity and numerical radius between semisimple Banach algebras. These results are useful in characterizations in quantum mechanics and operator algebras.
Sharon et al. (Tue,) studied this question.