Bivariant Hermitian K-theory and Karoubi's fundamental theorem

Let l be a commutative ring with involution & lowast; containing an element lambda such that lambda+lambda & lowast;=1 and let Alg(l)(& lowast;) be the category of l-algebras equipped with a semilinear involution and involution preserving homomorphisms. We construct a triangulated category kk(h) and a functor j(h):Alg(l)(& lowast;)-> kk(h) that is homotopy invariant, matricially and hermitian stable and excisive and is universal initial with these properties. We prove that a version of Karoubi's fundamental theorem holds in kk(h). By the universal property of the latter, this implies that any functor H:Alg(l)& lowast;-> T with values in a triangulated category which is homotopy invariant, matricially and hermitian stable and excisive satisfies the fundamental theorem. We also prove a bivariant version of Karoubi's 12-term exact sequence. (C) 2022 Elsevier B.V. All rights reserved.