In the present paper the real ideals of relatively compact operators of $W^*$-algebras are considered. Similar to the complex case, a description (up to isomorphism) of the real two-sided ideal of relatively compact operators of the complex $W^*$-factors is given.
"Real Compact Operators in Factors of Type I, II, and III$_\lambda$, $0 < \lambda < 1$." Missouri J. Math. Sci. 18 (1) 41 - 45, February 2006. https://doi.org/10.35834/2006/1801041