Kreĭn C*-categories

C*-categories are essentially norm-closed ∗-categories of bounded linear operators between Hilbert spaces. The purpose of this work is to identify suitable axioms defining Kreĭn C*-categories, i.e. those categories that play the role of C*-categories whenever Hilbert spaces are replaced by more general indefinite inner product Kreĭn spaces, and provide some basic examples. Finally we provide a Gel’fand-Naĭmark representation theorem for Kreĭn C*-algebras and Kreĭn C*-categories.