  We generalize the notion of nuclear maps from functional analysis by defining
nuclear ideals in tensored *-categories. The motivation for this study came
from attempts to generalize the structure of the category of relations to
handle what might be called ``probabilistic relations''. The compact closed
structure associated with the category of relations does not generalize
directly, instead one obtains nuclear ideals. We introduce the notion of
nuclear ideal to analyze these classes of morphisms. In compact closed
categories, we see that all morphisms are nuclear, and in the category of
Hilbert spaces, the nuclear morphisms are the Hilbert-Schmidt maps.
  We also introduce two new examples of tensored *-categories, in which
integration plays the role of composition. In the first, morphisms are a
special class of distributions, which we call tame distributions. We also
introduce a category of probabilistic relations which was the original
motivating example.
  Finally, we extend the recent work of Joyal, Street and Verity on traced
monoidal categories to this setting by introducing the notion of a trace ideal.
For a given symmetric monoidal category, it is not generally the case that
arbitrary endomorphisms can be assigned a trace. However, we can find ideals in
the category on which a trace can be defined satisfying equations analogous to
those of Joyal, Street and Verity. We establish a close correspondence between
nuclear ideals and trace ideals in a tensored *-category, suggested by the
correspondence between Hilbert-Schmidt operators and trace operators on a
Hilbert space.
