Bivariant $K$-theory and the Weyl algebra

We introduce a new version $k{k}^{\mathrm{a}\mathrm{l}\mathrm{g}}$ of bivariant $K$-theory that is defined on the category of all locally convex algebras. A motivating example is the Weyl algebra $W$, i.e. the algebra generated by two elements satisfying the Heisenberg commutation relation, with the fine locally convex topology. We determine its $k{k}^{\mathrm{a}\mathrm{l}\mathrm{g}}$-invariants using a natural extension for $W$. Using similar methods the $k{k}^{\mathrm{a}\mathrm{l}\mathrm{g}}$-invariants can be determined for many other algebras of similar type.