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kk$^{\text{ban}}$ is a bivariant K-theory for Banach algebras that hasreasonable homological properties, a product and is Morita invariant in a verygeneral sense. We define it here by a universal property and ensure itsexistence in a rather abstract manner using triangulated categories. Thedefinition ensures that there is a natural transformation from Lafforgue'stheory KK$^{\text{ban}}$ into it so that one can take products of elements inKK$^{\text{ban}}$ that lie in kk$^{\text{ban}}$.

kk-Theory for Banach Algebras I: The Non-Equivariant Case