Vector space isomorphisms of non-unital reduced Banach *-algebras

Rachid ElHarti, Mohamed Mabrouk
2015 Annales Universitatis Mariae Curie-Sklodowska. Sectio A. Mathematica  
Let A and B be two non-unital reduced Banach *-algebras and φ: A → B be a vector space isomorphism. The two following statement holds: If φ is a *-isomorphism, then φ is isometric (with respect to the C<sup>*</sup>-norms), bipositive and φ maps some approximate identity of A onto an approximate identity of B. Conversely, any two of the later three properties imply that φ is a *-isomorphism. Finally, we show that a unital and self-adjoint spectral isometry between semi-simple Hermitian Banach algebras is an *-isomorphism.
doi:10.17951/a.2015.69.2.61-68 fatcat:rod6wniotvbhnestb2ykadl5km