Real \(W^ *\)-algebras with an abelian Hermitian part. (Q1810094)

The present paper is devoted to the description of real \(W^{*}\)-algebras with abelian hermitian part. A real \(W^{*}\)-algebra is a real \(C^{*}\)-algebra \(\mathcal{R}\) such that \(\mathcal{R}+i\mathcal{R}\) is a complex \(W^{*}\)-algebra. The authors show that any real \(W^{*}\)-algebra with abelian hermitian part can be decomposed into the direct sum of two real \(W^{*}\)-algebras one of which is abelian and the other has a complex enveloping algebra of type \(I_2\). Then, showing that any real \(W^{*}\)-algebra with abelian hermitian part and a complex enveloping algebra of type \(I_2\) is \(*\)-isomorphic to \(L_{\mathbb{Q}}^\infty (\Omega ,\nu )\) (where \(\Omega \) is a hyper-Stonean compactum with Radon measure \(\nu \) and \(\mathbb{Q}\) is the skew field of quaternions) and using the fact that an abelian real \(W^{*}\)-algebra is \(*\)-isomorphic to a direct sum of algebras of the form \(L_{\mathbb{R}}^\infty (\Omega ,\nu )\) and \(L_{\mathbb{C}}^\infty (\Omega ,\nu )\), they obtain the description of real \(W^{*}\)-algebras with abelian hermitian part.