Lemma 12

Let A be a unital involutive algebra and let $x\in A$ be a hermitian element.

a) The element $x$ is positive if and only if $\|\|x\| \cdot 1-x\|\leqslant \|x\|$;

b) If $\|x\|\leqslant 1$, then $x$ is positive if and only if $\|1-x\|\leqslant 1$ ;

c) If $x$ is positive, then $1-x$ is positive if and only if $\|x\|\leqslant 1$ ;

d) If $x$ is positive and if $y\in A_+$ commutes with $x$, then $xy$ is positive.

The element $x$ is hermitian, hence normal. By considering the involutive subalgebra generated by $x$, which is commutative, one is reduced to the case where the algebra A is commutative, that is, to the case where A = $\mathscr{C}_0(X)$ for a locally compact topological space X (th. 1 of I, p. 108). The first three assertions then follow immediately from example 1 above. Analogously, to now prove assertion d), one may consider the involutive subalgebra generated by $x$ and $y$, which is commutative.