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Given a von Neumann algebra M, we consider the central extension E(M) of M. We introduce the topology t c(M) on E(M) generated by a center-valued norm and prove that it coincides with the topology of local convergence in measure on E(M) if and only if M does not have direct summands of type II. We also show that t c(M) restricted to the set E(M)h of self-adjoint elements of E(M) coincides with the order topology on E(M)h if and only if M is a σ-finite type Ifin von Neumann algebra.
The paper is devoted to a description of all real strongly facially symmetric spaces which are isometrically isomorphic to L₁-spaces. We prove that if Z is a real neutral strongly facially symmetric space such that every maximal geometric tripotent from the dual space of Z is unitary, then the space Z is isometrically isomorphic to the space L₁(Ω,Σ,μ), where (Ω,Σ,μ) is an appropriate measure space having the direct sum property.
Given a von Neumann algebra M we consider its central extension E(M). For type I von Neumann algebras, E(M) coincides with the algebra LS(M) of all locally measurable operators affiliated with M. In this case we show that an arbitrary automorphism T of E(M) can be decomposed as , where is an inner automorphism implemented by an element a ∈ E(M), and is a special automorphism generated by an automorphism ϕ of the center of E(M). In particular if M is of type then every band preserving automorphism...
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