  Given a II$_1$-factor $\mathcal{M}$ with tracial state $\tau$ and given an
$\mathcal{M}$-bimodule $\mathcal{E}(\mathcal{M},\tau)$ of operators affiliated
to $\mathcal{M}$ and a trace $\varphi$ on $\mathcal{E}(\mathcal{M},\tau)$,
(namely, a linear functional that is invariant under unitary conjugation), we
prove that $\det_\varphi:\mathcal{E}_{\log}(\mathcal{M},\tau)\to[0,\infty)$
defined by $\det_\varphi(T)=\exp(\varphi(\log |T|))$ is a multiplicative map on
the set $\mathcal{E}_{\log}(\mathcal{M},\tau)$ of all affiliated operators $T$
such that $\log_+(|T|)\in\mathcal{E}(\mathcal{M},\tau)$. Finally, we show that
all multiplicative maps on the invertible elements of
$\mathcal{E}_{\log}(\mathcal{M},\tau)$ arise in this fashion.
