| dc.description.abstract | Certain properties of operator algebras have been studied such as bound¬edness, positivity, surjectivity, linearity, invertibility, numerical range, nu¬merical radius and idempotent property. Of great interest is the study of spectrum of linear mappings. Jordan homomorphisms have been studied by several scholars. For instance,it has been shown that a linear map on two Banach algebras is a Jordan homomorphism and multiplicative. Fur¬thermore, Jordan homomorphisms between Von Neumann algebras have been shown to be spectrally bounded. However studies on spectral char¬acterization on semisimple Banach algebras have been done but to a little extent. It is therefore necessary to characterize Jordan homomorphisms on semisimple Banach algebras in terms of their spectrum. In this study we: Investigated whether Jordan homomorphisms on semisimple Banach algebras are spectral isometries; Investigated whether Jordan homomor¬phisms are unital surjections on semisimple Banach algebras and estab¬lished the relationship between unital surjections and spectral isometries on semsimple Banach algebras. In this study 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 it is also unital surjection if it preserves Jordan zero products and finally it 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. | en_US |