Journal of Operator Theory

Volume 78, Issue 1, Summer 2017 pp. 119-134.

Determinants associated to traces on operator bimodules

Authors: K. Dykema (1), F. Sukochev (2), and D. Zanin (3)
Author institution: (1) Department of Mathematics, Texas A and M University, College Station, TX 77843-3368, U.S.A.
(2) School of Mathematics and Statistics, University of New South Wales, Kensington, NSW 2052, Australia
(3) School of Mathematics and Statistics, University of New South Wales, Kensington, NSW 2052, Australia

Summary: Given a II$_1$-factor $\mathcal{M}$ with tracial state $\tau$ and given an $\mathcal{M}$-bi\-module $\mathcal{E}(\mathcal{M},\tau)$ of operators affiliated to $\mathcal{M}$ we show that traces on $\mathcal{E}(\mathcal{M},\tau)$ (namely, linear functionals that are invariant under unitary conjugation) are in bijective correspondence with rearrangement-invariant linear functionals on the corresponding symmetric function space $E$. We also show that, given a positive trace $\varphi$ on $\mathcal{E}(\mathcal{M},\tau)$, the map $\mathrm{det}_\varphi:\mathcal{E}_{\log}(\mathcal{M},\tau)\to[0,\infty)$ defined by $\mathrm{det}_\varphi(T)=\exp(\varphi(\log |T|))$ when $\log|T|\in\mathcal{E}(\mathcal{M},\tau)$ and $0$ otherwise, is multiplicative on the $*$-algebra $\mathcal{E}_{\log}(\mathcal{M},\tau)$ that consists 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.

DOI: http://dx.doi.org/10.7900/jot.2016may31.2123
Keywords: determinant, von Neumann algebra, {\rm II}$_1$-factor, noncommutative function space

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