We introduce and study the
concept of a bornological quantum group. This generalizes the theory of algebraic
quantum groups in the sense of van Daele from the algebraic setting to the
framework of bornological vector spaces. Working with bornological vector spaces
allows to extend the scope of the latter theory considerably. In particular, the
bornological theory covers smooth convolution algebras of arbitrary locally compact
groups and their duals. Another source of examples arises from deformation
quantization in the sense of Rieffel. Apart from describing these examples we obtain
some general results on bornological quantum groups. In particular, we construct
the dual of a bornological quantum group and prove the Pontrjagin duality
theorem.
