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Determinants associated to traces on operator bimodules. (English) Zbl 1389.46080
This important paper is a continuation of the authors’ works [Complex Anal. Oper. Theory 10, No. 8, 1775–1787 (2016; Zbl 1373.46059); Isr. J. Math. 222, No. 2, 645–709 (2017; Zbl 06821077)] and is largely influenced by the cited work of N. Karlton and F. Sukochev [Can. Math. Bull. 51, No. 1, 67–80 (2008; Zbl 1144.46054)] and the well-known paper of B. Fuglede and R. V. Kadison [Ann. Math. (2) 55, 520–530 (1952; Zbl 0046.33604)].
Two important results in this paper are Theorems 1.1 and 1.3. Theorem 1.1 is prompted by the work of Karlton and Sukochev [loc. cit.] and a major part of this theorem was already proved there, whereas Theorem 1.3, which is the main result of this paper, is prompted by the work of Fuglede et al. [loc. cit.]. Another noteworthy result is Proposition 1.7 which is based on a question of Amuthan Krishnaswamy-Usha, as a sequel to Theorem 1.3. The essence of both the theorems and the proposition are given in the paragraph below.
Let $$\mathcal{M}$$ be a von Neumann algebra factor of type $$\Pi_1$$ with tracial state $$\tau$$ and assume that $$\mathcal{M}$$ has a separable predual. Let $$E$$ be a Calkin function space and let $$\mathcal{E}(\mathcal{M}, \tau)$$ be the corresponding $$\mathcal{M}$$-bimodule. The authors show that there is a bijection from the set of all traces of $$\mathcal{E}(\mathcal{M}, \tau)$$ onto the set of all rearrangement invariant functionals of $$E$$, whereby a trace of $$\varphi$$ is mapped to a function $$\varphi_0$$ of $$E$$ satisfying $$\varphi_0 \left(\mu(A)\right)=\varphi(A)$$ where $$A \in \mathcal{E}(\mathcal{M}, \tau)$$ and $$A \geq 0$$. Theorem 1.3 shows that, given a positive trace $$\varphi$$ on $$\mathcal{E}(\mathcal{M}, \tau)$$, the map $$\operatorname{det}\varphi : \mathcal{E}_{\log}(\mathcal{M}, \tau) \to [0, \infty)$$ defined by $$\operatorname{det}\varphi(T)=\exp (\varphi(\log | T |))$$, where $$\log | T | \in \mathcal{E}(\mathcal{M}, \tau)$$ 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, the authors show that all multiplicative maps on the invertible elements of $$\mathcal{E}_{\log}(\mathcal{M}, \tau)$$ arise in this fashion.
To conclude, a special case of the proof of Theorem 1.3 yields an alternative proof of U. Haagerup and H. Schultz’s well-known result [Math. Scand. 100, No. 2, 209–263 (2007; Zbl 1168.46039)] about the extension of the Fuglede-Kadison determinant to $$\mathcal{L}_{\log}(\mathcal{M}, \tau)$$.

##### MSC:
 46L52 Noncommutative function spaces
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