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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