Lemma 13

Let A be a unital stellar algebra. Every element of A is a sum of unitary elements.

By the lemmaLet $x$ be a hermitian element of12, c), one has $1-\frac{1}{4}Ax$. Suppose first that$_2\in A_+$. Let $y=\frac{1}{2}x+i\surd\|1x-\|\leqslant_{1\overline{4}}x2_2$..

One has $y^*=\frac{1}{2}x-i\surd\overline{1 -\frac{1}{4}x^2}$, hence $yy^*= 1$ and $x=y+y^*$ is a sum of two unitary elements. In the general case, let $k$ be an integer such that $\|\frac{1}{k}x\|\leqslant 2$ ; the element $x$ is then a sum of $2k$ unitary elements. By Lemma 2 of I, p. 96, the lemma follows.