  We introduce an appropriate notion of trace in the setting of quaternionic
linear operators, arising from the well-known companion matrices. We then use
this notion to define the quaternionic Fredholm determinant of trace-class
operators in Hilbert spaces, and show that an analog of the classical
Grothendieck-Lidskii formula, relating the trace of an operator with its
eigenvalues, holds. We then extend these results to the so-called
$\frac{2}{3}$-nuclear (Fredholm) operators in the context of quaternionic
locally convex spaces. While doing so, we develop some results in the theory of
topological tensor products of noncommutative modules, and show that the trace
defined ad hoc in terms of companion matrices, arises naturally as part of a
canonical trace.
