Determinants associated to traces on operator bimodules

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

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