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Topologies on central extensions of von Neumann algebras

Shavkat Ayupov;  Karimbergen Kudaybergenov;  Rauaj Djumamuratov — 2012

Open Mathematics

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.

Automorphisms of central extensions of type I von Neumann algebras

Sergio Albeverio;  Shavkat Ayupov;  Karimbergen Kudaybergenov;  Rauaj Djumamuratov — 2011

Studia Mathematica

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 T = T a ∘ T ϕ , where T a ( x ) = a x a - 1 is an inner automorphism implemented by an element a ∈ E(M), and T ϕ is a special automorphism generated by an automorphism ϕ of the center of E(M). In particular if M is of type I ∞ then every band preserving automorphism...

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