Determinants associated to traces on operator bimodules
| Autor: | Dykema, K., Sukochev, F., Zanin, D. |
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| Rok vydání: | 2016 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 13 pages. Version 2 improves the statement and proof of Theorem 1.1 and adds Proposition 1.7 about arbitrary multiplicative maps |
| Databáze: | arXiv |
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