Proposition 15

Let A be a star-algebra. Let $x$ be a normal element of A.

a) Suppose that A is unital and let $f$ be a continuous function from Sp$_A(x)$ into $\mathbf{C}$. In order that $f(x)$ be positive, it is necessary and sufficient that the image of $f$ be contained in $\mathbf{R}_+$;

b) Let $f$ be a continuous function from Sp$'_A(x)$ into $\mathbf{C}$ such that $f(0) =$ 0. In order that $f(x)$ be a positive element of A, it is necessary and sufficient that the image of $f$ be contained in $\mathbf{R}_+$.

Assertion a) follows from assertion a) of cor. 2 of I, p. 111, and assertion b) follows from prop. 13 of I, p. 114.

Let $x$ be a hermitian element of the star-algebra A. Its spectrum is contained in $\mathbf{R}$ (prop. 4 of I, p. 106). Consider the continuous functions from Sp$'_A(x)$ into $\mathbf{R}$ defined by

$f_1:t\mapsto$ sup($t,0$)$,f_2:t\mapsto$ sup($-t,0$)$,f_3:t\mapsto |t|$.

We put

Since the functions $f_1,f_2,f_3$ take positive real values and vanish at 0, the elements $x^+,x^-$ and $|x|$ are positive elements of A (Prop. 15, a)) which belong to the stellar subalgebra of A generated by $x$.

One has $f_1(t)-f_2(t) =t$ for every $t\in \mathbf{R}$, as well as the relations $f_1+f_2=f_3$ and $f_1f_2= 0$. Hence the relations:

Since the functional calculus mapping is isometric, one has

Let $x$ be a positive element of A. It is hermitian, hence normal. Let $\alpha \in \mathbf{R}^*_+$, and let $g$ be the restriction to Sp$'_A(x)$ of the function $t\mapsto t^{\alpha}$; one writes $x^{\alpha}=g(x)$. It is a positive element of the stellar subalgebra of A generated by $x$. Let $\alpha$ and $\beta$ be in $\mathbf{R}^*_+$. Since the functional calculus mapping is an algebra morphism, and by Prop. 13 of I, p. 114, one has

One shall also write $\overline{x}=x^{12}$.