Abstract
<jats:p>The paper studies quasitraces on real C*-algebras, and AW*-completion of C*-subalgebras with respect to the $d_\tau$-metric generated by quasitrace $\tau$. It is proved that the $d_\tau$-closure of unital real C*-subalgebra $B$ of real C*-algebra $R$ is the smallest real AW*-subalgebra of $R$ containing $B$. To prove this, it was necessary to obtain a key result concerning the maximal Abelian self-adjoint subalgebra (masa), in connection with which Abelian algebras are studied separately. It is proved that for a compact Hausdorff space $X$ the algebra $C_r(X)$ of all continuous real functions on $X$ is a real abelian AW*-algebra if and only if $X$ is Stonean. Moreover, it has been proven that a unital real C*-algebra is a real AW*-algebra if and only if every masa has Stonean spectrum, and this is equivalent to the fact that every masa is monotone complete.</jats:p>