kk-Theory for Banach algebras I: The non-equivariant case

kkban is a bivariant K-theory for Banach algebras that has reasonable homological properties, a product and is Morita invariant in a very general sense. We define it here by a universal property and ensure its existence in a rather abstract manner using triangulated categories. The definition ensures that there is a natural transformation from Lafforgue's theory KKban into it so that one can take products of elements in KKban that lie in kkban.