Determinants associated to traces on operator bimodules

TL;DR

This paper characterizes traces on operator bimodules affiliated with II$_1$-factors, establishing a multiplicative determinant function and classifying all such multiplicative maps on invertible elements.

Contribution

It introduces a trace-based determinant function on operator bimodules and classifies all multiplicative maps on their invertible elements.

Findings

The determinant function $ ext{det}_ au$ is multiplicative on a specific class of affiliated operators.

All multiplicative maps on invertible elements of the bimodule are characterized by this determinant.

The work extends the understanding of traces and determinants in the context of operator algebras.

Abstract

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.