Index for von Neumann algebras with finite dimensional centers
Let \(M\) and \(N\) be von Neumann algebras and let \(E\in {\mathcal E}(M, N)\), the set of normal faithful conditional expectations from \(M\) onto \(N\). \textit{H. Kosaki} [J. Funct. Anal. 66, 123-140 (1986; Zbl 0607.46034)] extended the Jones' index theory to pairs of arbitrary factors \(N\subset M\) defining Index \(E\), for \(E\in {\mathcal E}(M, N)\), to be \(E^{-1}(1)\), where the operator-valued weight \(E^{- 1}\) [\textit{U. Haagerup}, J. Funct. Anal. 32, 175-206 (1979; Zbl 0426.46046)] is defined using \textit{A. Connes'} spatial Radon-Nikodym derivatives [J. Funct. Anal. 35, 153-164 (1980; Zbl 0443.46042)]. The author allows \(M\) and \(N\) to be von Neumann algebras with finite-dimensional centres. Kosaki's Index \(E\) is now a positive operator affiliated to \(Z(M)\), the centre of \(M\). When there exists an \(E\in {\mathcal E}(M, N)\) such that Index \(E\) is bounded, \(N\) is said to be of finite index in \(M\). When \(N\) is not of finite index the author sets the `index' \([M; N]_0\) to be \(\infty\) (see below). \textit{H. Araki} [Publ. Res. Ind. Math. Sci., Kyoto Univ. 11, 809-833 (1976; Zbl 0326.46031)] using the modular operator of \textit{M. Takesaki} and Tomita [`Tomita's theory of modular Hilbert algebras and its applications', Lect. Notes Math. 128 (1970; Zbl 0193.425)] extended the relative entropy for density matrices to faithful normal states of von Neumann algebras. (Intuitively the relative entropy is the logarithm of the number of intermediate states.) Supposing that there exists a faithful normal state \(\varphi\) of \(M\) for which there exists an \(E\in {\mathcal E}(M, N)\) with \(\varphi\circ E= \varphi\), the author adapts Araki to define the entropy \(K_\varphi(M|N)\) of \(M\) relative to \(\varphi\) and \(N\). \(K_E\) denotes the supremum of the \(K_\varphi\) such that \(\varphi\circ E= \varphi\). Assume henceforth that \(Z(M)\cap Z(N)= \mathbb{C}\) and \(E\in {\mathcal E}(M, N)\). If \(N\) is of finite index in \(M\) then there exists a unique \(E_0\in {\mathcal E}(M, N)\) such that \(|\text{Index }E_0|\) is the minimum \(|\text{Index }E|\). Index \(E_0\) is shown to be a scalar, denoted by \([M; N]_0\), and it is called the minimum index. The author, following \textit{F. Hiai's} [Publ. Res. Inst. Math. Sci. 24, No. 4, 673-678 (1988; Zbl 0679.46050)] result for factors, shows that \(E= E_0\) if and only if \(K_E(M|N)= \log|\text{Index } E|\). Another necessary and sufficient condition concerns \(E\) determined by the \textit{J.-F. Havet} [J. Oper. Theory 24, No. 1, 33-55 (1990; Zbl 0782.46051)] decomposition. For central projections \((p_i)\) of \(M\) and \((q_j)\) of \(N\), central decompositions \(M_{p_i q_j}\) of \(M\), \(N_{p_i q_j}\) of \(N\) and \(E_{ij}\in E\in {\mathcal E}(M_{p_i q_j}, N_{p_i q_j})\) have been defined by Havet (loc. cit.). If the decomposition and `stochastic' \(n\times m\) matrices determine \(E\) in the sense of Havet (loc. cit.) then Index \(E\) is scalar and \(|\text{Index } E_{ij}|= [M; N]_0\); an additional condition on the decomposition also assures the sufficiency. As in \textit{F. Hiai} [J. Oper. Theory 24, No. 2, 301-336 (1990; Zbl 0804.46073)] the index is shown to be well behaved under taking tensor products and crossed products by finite groups.
Connes' spatial Radon-Nikodym derivatives
normal faithful conditional expectations
Jones' index theory to pairs of arbitrary factors
faithful normal states of von Neumann algebras
scientific article; zbMATH DE number 597440
scientific article