Determinants and the formula "det(exp(y)) = exp(trace(y))", theme and variation

One of the possible definitions of the determinant of a complex invertible matrix is the formula $\det(\exp(y))=\exp(\mathrm{trace}(y))$. This is the starting point of the determinant of Fuglede and Kadison (1952, in the setting of operator algebras), that we will revisit. We will explain how this can be adapted to other situations, and analyse why, depending on the setting, it provides a determinant which can be real-valued, or complex-valued, or indeed with other values. We will briefly discuss the relevance of this to K-theory and topology (torsion).