A bicommutant theorem for completely bounded module homomorphisms
Throughout suppose that \(H\) is a separable Hilbert space and \(R\) is a von Neumann algebra in \(B(H)\). Denote by \(E=EL_{R'}(B(H))\) the elementary \(R'\)-bimodule homomorphisms of \(B(H)\). The relative commutant of \(E\) in \(CB(B(H))\), the completely bounded operators on \(B(H)\), is the space of completely bounded \(R\)-bimodule homomorphisms of \(B(H)\). The main result of this paper is that the bicommutant of \(E\) in \(CB(B(H))\) is the subalgebra of normal \(R'\)-bimodule homomorphisms provided that \(R\) has no atomic, central direct summands of type \(I_{\infty,n}\) for any natural number \(n\). This is proved by showing that, under the given assumptions, any linear \(R'\)-bimodule homomorphism which commutes with every completely bounded \(R\)-bimodule homomorphism is bounded and normal. The authors also prove some results on commutants in the linear operators on \(B(H)\). They show that if \(R\) is maximal commutative then the bicommutant of \(E\) is the algebra of all normal completely bounded \(R'\)-bimodule homomorphisms of \(B(H)\). From this it follows that if \(R\) is commutative then the centre of the completely bounded \(R'\) homomorphisms of \(B(H)\) is the subalgebra of normal operators. A partial result is also obtained under the assumption that \(R\) is continuous, namely that every linear operator on \(B(H)\) which commutes with all completely bounded \(R\)-bimodule homomorphisms is a normal completely bounded \(R'\)-bimodule homomorphism. The main tool in their study is the multilinear double commutant theorem of \textit{E. G. Effros} and \textit{R. Exel} [Lond. Math. Soc. Lect. Note Ser. 135, 81-94 (1988; Zbl 0698.46049)]which says that the relative commutant of the normal \(R'\)-bimodule homomorphisms of \(B(H)\) in the completely bounded operators is the space of completely bounded \(R\)-bimodule homomorphisms of \(B(H)\).
scientific article; zbMATH DE number 1031504