O‘zbekiston matematika jurnali Том 70 № 1 (2026)
Normality of quasitrace and AW*-completion of the real C*-subalgebras
Rakhmonova, N.
Аннотация
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.
{Real C*-algebrasAW*-algebrasquasitracemonotone completenessmaximal abelian self-adjoint subalgebra (masa).}
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