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Brown measures of unbounded operators affiliated with a finite von Neumann algebra. (English) Zbl 1168.46039
The purpose of this paper is to generalize Brown’s spectral distribution measure to a large class of unbounded operators affiliated with a finite von Neumann algebra. Let \({\mathcal M}\) be a finite von Neumann algebra with a faithful normal tracial state \(\tau\), and let \[ \varDelta (T) = \exp \left( \int_0^{\infty} \log t \text{d} \mu_{ | T | } (t) \right) \tag{1} \] denote the corresponding Fuglede–Kadison determinant [cf. B. Fuglede and R. V. Kadison, Ann. Math. (2) 55, 520–530 (1952; Zbl 0046.33604)]. [L. G. Brown, Pitman Res. Notes Math. Ser. 123, 1–35 (1986; Zbl 0646.46058)] proved the unique existence of a compactly supported probability measure \(\mu_T\) on \({\mathbb C}\) for every \(T\in{\mathcal M}\), such that \[ \log\varDelta (T-\lambda 1)=\int_{{\mathbb C}}\log| z-\lambda|\text{d}\mu_T (z)\quad\text{for}\quad\lambda\in{\mathbb C}.\tag{2} \] This measure \(\mu_T\) is called Brown’s spectral distribution measure (or simply the Brown measure) of \(T\). For \(\mu \in\) \(\text{Prob}( (0, \infty))\), define \(\psi_{\mu}\) : \({\mathbb C} \setminus (0, \infty)\) \(\to\) \({\mathbb C}\) by \[ \psi_{\mu}(z) = \int_0^{\infty} \frac{ \text{d} \mu(t)}{1 - z t} - 1 \quad \text{for} \quad z \in {\mathbb C} \setminus (0, \infty), \tag{3} \] and notice that \(\psi_{\mu}\) maps a connected neighborhood \({\mathcal U}(\mu)\) of \((-\infty, 0)\) injectively onto a neighborhood \({\mathcal V}(\mu)\) of \((-1, 0)\). Then the \(S\)-transform \({\mathcal S}_{\mu}:{\mathcal V}(\mu)\to{\mathbb C}\) of the probability measure \(\mu\) on \((0, \infty)\) is defined as \[ {\mathcal S}_{\mu}(z)=\frac{z+1}{z}\psi_{\mu}^{-1}(z), \quad z \in {\mathcal V}(\mu).\tag{4} \] Indeed, the map \(\mu\) \(\mapsto\) \({\mathcal S}_{\mu}\) is one-to-one on \(\text{Prob}( (0, \infty))\). It was proven by V. Haagerup and F. Larsen [J. Funct. Anal. 176, No. 2, 331–367 (2000; Zbl 0984.46042)] that, if \(T \in {\mathcal M}\) is \(R\)-diagonal in the sense of [A. Nica and R. Speicher, Fields Inst. Commun. 12, 149–188 (1997; Zbl 0889.46053)], then \(\mu_T\) can be determined from the \(S\)-transform of the distribution \(\mu_{ | T |^2}\). In fact, as to the case of \(T \in {\mathcal M}\) where \(T\) is an \(R\)-diagonal element and not proportional to a unitary and \(\text{ker}(T)\) \(= 0\), \(\mu_T\) is the unique probability measure on \({\mathbb C}\) which is invariant under the rotations \(z \mapsto \gamma z\) for \(\gamma \in T\), and which satisfies \[ \mu_T\left\{B\left(0,{\mathcal S}_{\mu_{| T|^2}}(t-1)^{- 1/2}\right)\,\right\}=t,\quad 0<t <1.\tag{5} \] In the paper under review, the authors extend the Brown measure to all operators in the set \({\mathcal M}^{\varDelta}\) of closed densely defined operators \(T\) affiliated with \({\mathcal M}\) satisfying \[ \int_0^{\infty}\log^+ t\,\text{d}\mu_{| T|}(t)<\infty, \tag{6} \] where \(\log^+ t= \max\{\log t,0\}\). Moreover, the authors extend their previous result to all unbounded \(R\)-diagonal operators in the class \({\mathcal M}^{\varDelta}\) and also compute the Brown measures of all \(T\in{\mathcal M}^{\varDelta}\). Finally, they study a particular example of an unbounded \(R\)-diagonal element, namely, the operator \(z = x y^{-1}\), where \((x, y)\) is a circular system in the sense of [V. D. Voiculescu, Progr. Math. 92, 45–60 (1990; Zbl 0744.46055)], and prove that the Brown measure of \(z\) is given by \[ \text{d} \mu_z(s) = \frac{1}{ \pi ( 1 + | s |^2)^2} \text{d} ( {\mathcal R}e \, s) \text{d} ({\mathcal I}m \, s), \tag{7} \] and also that \(z^n \in L^p({\mathcal M}, \tau)\) if and only if \(0 < p < 2/(n+1)\).

MSC:
46L54 Free probability and free operator algebras
46L10 General theory of von Neumann algebras
47A15 Invariant subspaces of linear operators
47B99 Special classes of linear operators
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