Lemma 15

Let A be a stellar algebra and let I be a closed two-sided ideal of A. Then I is self-adjoint.

Let $J = I\cap I^*$. The set J is a self-adjoint two-sided ideal of A, which contains $I^*I$. In particular, J is a stellar algebra. Let $\mathfrak{F}$ be an increasing approximate unit of J (Prop. 18 of I, p. 121). For every $x\in I$ and every $f\in J^{\leqslant 1}_+$, one has

Since $x^*x\in J$, one has therefore

lim$_{f,\mathfrak{F}}\|xf-x\|^2= 0$.

Since $xf\in J$ for every $f\in J^{\leqslant 1}_+$ and since J is closed, this implies that $x\in J$. Hence I = J, and the ideal I is self-adjoint.