On von Neumann algebras consisting of complex symmetric operators

An operator $ T \in B(\mathcal {H}) $ T is an element of B(H) is said to be complex symmetric if there exists a conjugate-linear, isometric involution $ C: \mathcal {H} \rightarrow \mathcal {H} $ C:H -> H so that $ T=C T<^>* C $ T=CT & lowast;C. We prove that a von Neumann algebra acting on a separable Hilbert space consists of complex symmetric operators if and only if it is unitarily equivalent to a direct sum of (some of the summands may be absent): (i) an abelian von Noumann algebra. (ii) a type $ \mathrm {I}_2 $ I2 von Neumann algebra.