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Proper analytic free maps. (English) Zbl 1215.47011

The paper under review concerns analytic free maps. These are the free analogues of classical analytic functions in several complex variables and are defined in terms of several complex variables. To give the precise definitions requires some preliminary notation.
Fix a positive integer \(g\). Given a positive integer \(n\), let \(M_n(\mathbb{C)}^g\) denote \(g\)-tuples of \(n\times n\) matrices. A sequence \(\mathcal{U}=(\mathcal{U} (n))_{n\in\mathbb{N}},\) where \(\mathcal{U} (n) \subset M_n(\mathbb{C)}^g\), is a noncommutative set if it is closed with respect to simultaneous unitary similarity; i.e., if \(X\in \mathcal{U} (n)\) and \(U\) is an \(n\times n\) unitary matrix, then
\[ U^\ast X U =(U^\ast X_1U,\dots, U^\ast X_gU)\in \mathcal{U} (n); \] and if it is closed with respect to direct sums; i.e., if \(X\in \mathcal{U} (n)\) and \(Y\in \mathcal{U}(m)\) implies
\[ X\oplus Y = \begin{pmatrix} X & 0\\ 0 & Y \end{pmatrix} \in \mathcal{U}(n+m). \]
The noncommutative set \(\mathcal{U}\) is a noncommutative domain if each \(\mathcal{U}(n)\) is open and connected. The noncommutative set \(\mathcal{U}\) is bounded if there is a \(C\in\mathbb R\) such that
\[ C^2 -\sum X_j X_j^\ast \succ 0 \]
for every \(n\) and \(X\in\mathcal{U}(n)\).
Let \(\mathcal{U}\) denote a noncommutative subset of \(M_n(\mathbb{C)}^g\) and let \(\tilde{g}\) be a positive integer. A free map \(f\) from \(\mathcal{U}\) into \(M_n(\mathbb{C)}^{\tilde{g}}\) is a sequence of functions \(f[n]:\mathcal{U}(n) \to M_n(\mathbb{C)}^{\tilde{g}}\) which respects intertwining maps; i.e., if \(X\in\mathcal{U}(n)\), \(Y\in\mathcal{U}(m)\), \(\Gamma:\mathbb C^m\to\mathbb C^n\), and
\[ X\Gamma=(X_1\Gamma,\dots, X_g\Gamma)=(\Gamma Y_1,\dots, \Gamma Y_g)=\Gamma Y, \]
then \(f[n](X) \Gamma = \Gamma f[m] (Y)\). An alternate characterization of free maps is that they are the maps that respect similarity and direct sums.
Given noncommutative domains \(\mathcal{U}\) and \(\mathcal{V}\) in \(M_n(\mathbb{C)}^g\) and \(M_n(\mathbb{C)}^{\tilde{g}}\), respectively, a free map \(f:\mathcal{U}\to\mathcal{V}\) is proper if each \(f[n]:\mathcal{U}(n)\to \mathcal{V}(n)\) is proper in the sense that, if \(K\subset \mathcal{V}(n)\) is compact, then \(f^{-1}(K)\) is compact. In particular, for all \(n\), if \((z_j)\) is a sequence in \(\mathcal{U}(n)\) and \(z_j\to\partial\mathcal{U}(n)\), then \(f(z_j)\to\partial\mathcal{V}(n)\). In the case where \(g=\tilde{g}\) and both \(f\) and \(f^{-1}\) are (proper) analytic free maps, \(f\) is a called a bianalytic free map. The following theorem is the central result of this paper:
Theorem. Let \(\mathcal{U}\) and \(\mathcal{V}\) be noncommutative domains containing \(0\) in \(M_n(\mathbb{C)}^g\) and \(M_n(\mathbb{C)}^{\tilde{g}}\), respectively, and suppose that \(f:\mathcal{U}\to \mathcal{V}\) is a free map. Then 6.5mm
(1)
If \(f\) is proper, then it is one-to-one, and \(f^{-1}:f(\mathcal{U})\to \mathcal{U}\) is a free map.
(2)
If, for each \(n\) and \(Z\in M_n(\mathbb{C)}^{\tilde{g}}\), the set \(f[n]^{-1}(\{Z\})\) has compact closure in \(\mathcal{U}\), then \(f\) is one-to-one and, moreover, \(f^{-1}:f(\mathcal{U})\to \mathcal{U}\) is a free map.
(3)
If \(g=\tilde{g}\) and \(f:\mathcal{U}\to\mathcal{V}\) is proper and continuous, then \(f\) is bianalytic.
It is important to note that the conditions imposed on the map \(f\) are sharp. The authors then proceed to use their main theorem to study analogues of classical theorems in several complex variables.
The paper concludes with examples of these maps in one variable. These examples are constructed by using linear matrix inequalities.

MSC:

47A13 Several-variable operator theory (spectral, Fredholm, etc.)
46L54 Free probability and free operator algebras
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References:

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