Surjective isometries of C*-algebras: complex spectrum and periodicity

Abstract In this paper, we are interested in studying surjective isometries of C*-algebras with an emphasis on their complex spectrum. We do not require the isometries to be linear nor the C*-algebras to be unital. We first characterize such isometries in terms of a Jordan ∗-isomorphism, a central projection, and a unitary element of the multiplier algebra, following a long line of work that began with the Banach-Stone theorem. In our main result, we then establish, for a wide class of surjective isometries, a precise connection between the complex spectrum of the isometry and the classical spectrum of the associated Jordan ∗-isomorphism. Finally, we turn our attention to periodic surjective isometries and provide several examples that illustrate the range of possibilities that can occur for the complex spectrum of the isometry and classical spectrum of the Jordan ∗-isomorphism.