Derivations of von Neumann algebras into the compact ideal space of a semifinite algebra
Let M be a semifinite von Neumann algebra, \({\mathcal J}(M)\) the closed ideal generated by the finite projection and N a weakly closed *-subalgebra of M. N is decomposed as \(N=N_ 0\oplus N_ 1\), where \(N_ 0\) is a finite type I von Neumann algebra and \(N_ 1\) a von Neumann algebra with no finite type I summands. One of the main results is that every derivation of N into \({\mathcal J}(M)\) is inner, i.e. of the form \(\delta (x)=Kx-xK\) for some \(K\in {\mathcal J}(M)\), provided that there exists a partition of the unity \(\{p_ i\}\) in the center \({\mathcal Z}(N_ 0)\) of \(N_ 0\) such that for each i it holds that \({\mathcal Z}(N_ 0)p_ i\subset {\mathcal Z}(M)p_ i\) or \({\mathcal Z}(M)p_ i\subset {\mathcal Z}(N_ 0)p_ i.\) The other main result is the following: if \(M=L^{\infty}[0,1]{\bar \oplus}{\mathcal B}(L^ 2({\mathbb{T}}))\) and \(N=1\oplus L^{\infty}({\mathbb{T}})\subset M\), there exists an operator \(T\in M\) which commutes modulo \({\mathcal J}(M)\) with all the elements of N but which is not a \({\mathcal J}(M)\)-permutation of an element commuting with M. In particular ad T is a not inner derivation from M into \({\mathcal J}(M)\).
von Neumann algebra with no finite type I summands
scientific article; zbMATH DE number 4101893