Inclusions of second quantization algebras (Q2702386)

Let \({\mathcal H}\) be a separable complex Hilbert space, and \(M_0\), \(M_1\) the closed real linear subspaces of \({\mathcal H}\). The symbol \({\mathcal R}(M_i)\) \((i=0,1)\) denotes the von Neumann algebra on the Fock space \(e^{{\mathcal H}}\) generated by the Weyl unitaries with test functions in \(M_i\). When \(K\) is a closed real linear subspace of \({\mathcal H}\), we write \(K \leq_R {\mathcal H}\). A pair \((E,F)\) with \(E, F\) \(\leq_R\) \({\mathcal H}\) is said to be a standard pair if the intersection \(E \wedge F\) \(=\) \(\{ 0 \}\) and the generated real subspace \(E \vee F\) \(=\) \({\mathcal H}\). On the assumption that \({\mathcal R}(M_0)\) and \({\mathcal R}(M_1)\) are standard with respect to the vacuum vector \(e^0 \in\) \(e^{{\mathcal H}}\), the authors study inclusions \({\mathcal R}(M_0)\) \(\subset\) \({\mathcal R}(M_1)\). Then the tower and the tunnel associated with the above inclusion is considered with the Tomita conjugation. \({\mathcal R}(M_0)\) \(\subset\) \({\mathcal R}(M_1)\) is said of depth 2 if \({\mathcal R}(M_0)' \cap {\mathcal R}(M_3)\) is a factor. The authors show first that \({\mathcal R}(M_0)\) \(\subset\) \({\mathcal R}(M_1)\) is of depth 2 if \(M_0 \subset M_1\) is irreducible, and also that for a not necessarily irreducible inclusion \(M_0 \subset M_1\) of standard subspaces with codimension \(n\), \({\mathcal R}(M_1)\) is isomorphic to the cross product of \({\mathcal R}(M_0)\) with \(R^n\), i.e., \({\mathcal R}(M_1)\) \(=\) \({\mathcal R}(M_0)\) \(\times_{\alpha} R^n\). The proof is due to the \textit{M. Landstad} theorem [Trans. Am. Math. Soc. 248, 223-267 (1979; Zbl 0397.46059)]. Moreover, they prove that there exists an irreducible inclusion \(M_0 \subset M_1\) of standard subspaces with infinite codimension for which the third relative commutant is of type III. The proof is partly based on the previous result: \textit{F. Figliolini} and \textit{D. Guido} [J. Operat. Theory 31, No. 2, 229-252 (1994; Zbl 0847.46042)].NEWLINENEWLINEFor the entire collection see [Zbl 0953.00049].