Determinants associated to traces on operator bimodules

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 $\phi$ on $\mathcal{E}(\mathcal{M},\tau )$, (namely, a linear functional that is invariant under unitary conjugation), we prove that ${\det }_{\phi }:{\mathcal{E}}_{\log }(\mathcal{M},\tau )\to [0,\infty )$ defined by ${\det }_{\phi }(T)=\exp (\phi (\log |T|))$ 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.

13 pages. Version 2 improves the statement and proof of Theorem 1.1 and adds Proposition 1.7 about arbitrary multiplicative maps

References in corpus (2)

• Algebras of log-integrable functions and operators
• An upper triangular decomposition theorem for some unbounded operators affiliated to II_1-factors