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The von Neumann algebras generated by \(t\)-Gaussians. (English) Zbl 1116.46056

Let \(H\) be a Hilbert space, which is the complexification of a real Hilbert space \(H_{{\mathbb R}} \), \(e_{i}\), \(i=1,\dots\) be an orthonormal basis in \(H_{{\mathbb R}}\) and let \(t>0\). The \(t\)-deformed Fock space is \({\mathcal F}_{t}= {\mathbb C}\Omega\oplus_{k\geq 1}t^{k-1}\otimes^{k}H\), where \(t^{k-1}\) means the multiplication of the scalar product by \(t^{k-1}\). For \(e\in H_{{\mathbb R}}\), \(l_{t}(e)\) is defined by \(l_{t}(e)\Omega =e\), \(l_{t}(e)(h_{1}\otimes\cdots\otimes h_{k})= e\otimes h_{1}\otimes\cdots\otimes h_{k}\); the \(t\)-Gaussian is \(s^{t}(e)=l_{t}(e)+l_{t}(e)^{\ast}\); for \(n=1\), \(s^{t}(1)\) is denoted by \(s^{t}\). If \(n=\dim H\), let \(\Gamma_{t,n}\) and \(C_{t,n}\) denote the von Neumann algebra (C\(^{\ast}\)-algebra) generated by all \(l_{t}(e)\), \(e\in H_{{\mathbb R}} \). Let \(\varphi =\langle \cdot\Omega ,\Omega \rangle\); this is a trace on \(\Gamma_{1,n}\). If \(n<\infty\) and \(t\notin [n(n+n^{1/2})^{-1},n( n-n^{1/2})^{-1}]\), then \(\Gamma_{t,n}=B(l_{2})\oplus\Gamma_{1,n}\) (and \(\varphi\) is not faithful on \(\Gamma_{t,n}\)); if \(t\in [n(n+n^{1/2})^{-1},n(n-n^{1/2})^{-1}]\), then \(\Gamma_{t,n}\) is isomorphic to \(\Gamma_{1,n} \), and \(C_{t,n}\) to \(C_{1,n}\). If \(n=\infty\), then \(\Gamma_{t,n}=B({\mathcal F}_{t})\).
On the way to these results, \(\pi_{i}:\Gamma_{t,1}\to\Gamma_{t,n}\), defined by \(\pi_{i}( s^{t})=s^{t}( e_{i})\), \(\varrho :\Gamma_{t,1}\to\Gamma_{1,1}\), defined by \(\varrho (s^{t})=t^{1/2}s^{1}\) (normal \(\ast \)-representations), and \(\psi_{i}=\varphi\circ\varrho\circ\pi_{i}^{-1}\) on \(A_{i}= \pi_{i}(\Gamma_{t,1})\) are considered. \((A_{i},\varphi ,\psi_{i})\) are conditionally free, i.e., \(\varphi (a_{1}\cdots a_{p})= \varphi (a_{1})\cdots\varphi (a_{p})\) for \(a_{j}\in A_{i_{j}}\), \(i_{j}\neq i_{j+1}\), \(\psi_{i_{j}}(a_{j})=0\). In a general situation as such, where \(A_{i}=L_{\infty}(I_{i},\mu_{i})\), \(\varphi (g)=\int g\,d\mu_{i}\) on \(A_{i}\) and \(\psi (g)=\int gf_{i}\,d\mu_{i}\), with bounded \(f_{i}\) and the \(\varphi\)-distribution of \(1+\sum_{i=1}^{n}(f_{i}-1)\) has no atom at \(0\), the author shows that the von Neumann algebra of the GNS of the conditionally free product is the free product of those of \((A_{i},\psi_{i})\). \(\varrho\) appears as the Calkin map and generalizes to a surjective normal \(\ast \)-representation \(\Gamma_{t,n}\to\Gamma_{t,1}\).

MSC:

46L54 Free probability and free operator algebras
46L10 General theory of von Neumann algebras
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References:

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