부산대학교 도서관

An operator $\varepsilon$ in a $C^*$-algebra $A$ is called a reflection if $\varepsilon^2=1$. If $\varepsilon=\mu\rho$ is the polar decomposition of $\varepsilon$, $\mu>0$, $\rho$ unitary, the authors show, under the implicit assumption that $A$ contains all spectral projections of $\mu$, that $\rho$ is the only unitary reflection in $A$ which satisfies the following. For each selfadjoint projection $p\in A$ that commutes with $\varepsilon$, $\rho$ satisfies the following conditions: (i) $\rho$ commutes with $p$; (ii) the reduction of $\rho$ to $A_p=pAp$ minimizes the distance from the reduction of $\varepsilon$ to $A_p$ to the set of unitary reflections in $A_p$.