Determinants associated to traces on operator bimodules.

*(English)*Zbl 1389.46080This 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 \(\mathcal{L}_{\log}(\mathcal{M}, \tau)\).

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 \(\mathcal{L}_{\log}(\mathcal{M}, \tau)\).

Reviewer: Madathum K. Viswanath (Chennai)

##### MSC:

46L52 | Noncommutative function spaces |