Skew commutators and Lie isomorphisms in real von Neumann algebras (Q1915347)

Let \(R\) be a real von Neumann algebra. Then its selfadjoint elements \(R_s= \{x\in R, x^*= x\}\) form a weakly closed Jordan algebra, i.e., JW-algebra with respect to symmetrized product \(x\circ y= xy+ yx\), while its skew-adjoint elements \(R_k= \{x\in R, x^*= -x\}\) form a real Lie algebra with brackets \([x,y]= xy- yx\). It was proved by the author in an earlier paper that if \(R\) and \(S\) are real factors nonisomorphic to \(\mathbb{R}\), then the JW-algebras \(R_s\) and \(S_s\) are isomorphic if and only if \(R\) and \(S\) are (associative) \(*\)-isomorphic. In the present paper, the author considers a similar problem for Lie algebras \(R_k\) and \(S_k\). Denote \([R_k, R_k]\) the derived Lie ring, i.e., the additive span of all commutators \([x,y]\), where \(x\), \(y\in R_k\); it is a Lie ideal in the Lie ring \(R_k\). The main result states that if \(R\) and \(S\) are real factors not of type \(I_1\) and \(I_2\) then \([R_k, R_k]\) and \([S_k, S_k]\) are isomorphic if and only if \(R\) and \(S\) are \(*\)-isomorphic. In particular, \(R_k\) and \(S_k\) are Lie isomorphic if and only if \(R\) and \(S\) are \(*\)-isomorphic. Also, an application to Lie algebras of form \(M^\alpha(- 1)= \{x\in M, \alpha(x)= -x\}\) is given, where \(\alpha\) is a \(*\)-anti-automorphism of order 2 of a (complex) von Neumann algebra \(M\).