Geodesic
On the coincidence of types of a~real $AW^*$-algebra and its complexification
Izvestiya. Mathematics , Tome 68 (2004) no. 5, pp. 851-860.

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We consider real $AW^*$-algebras, that is, Kaplansky algebras over the field of real numbers. As in the case of complex von Neumann algebras and complex $AW^*$-algebras, real $AW^*$-algebras are classified in terms of types $\mathrm{I}_{\mathrm{fin}}$, $\mathrm{I}_\infty$, $\mathrm{II}_1$, $\mathrm{II}_\infty$, and $\mathrm{III}$. We prove that if the complexification $M=A+iA$ of a real $AW^*$-algebra A also is an $AW^*$-algebra, then the types of $A$ and $M$ coincide. 
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S. A. Albeverio; Sh. A. Ayupov; A. Kh. Abduvaitov. On the coincidence of types of a~real $AW^*$-algebra and its complexification. Izvestiya. Mathematics , Tome 68 (2004) no. 5, pp. 851-860. http://geodesic.mathdoc.fr/item/IM2_2004_68_5_a0/

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