An Order-Preserving Linear Map from Matrices to Banach *-Algebras and Applications

We establish a theorem for which a number of absolute-valueidentities and inequalities in the framework of Banach -algebras can begenerated. To do this job, we construct anorder-preserving linear map from the vector space of 2-by-2 hermitian matrices to ahermitian Banach -algebras. This map can convert any suitable matrix orderingto a number of identities and inequalities in Banach -algebras. Hence,we obtain a number of analogues of the well-known results in a framework ofhermitian Banach -algebras