Proposition 11

Let A be a unital involutive algebra, and let $a,x$ and $y$ be elements of A. Suppose that $x$ and $y$ are normal. If $xa=$ $ay$, then one has $f(x)a=af(y)$ for every function $f$ continuous in the union of the spectrum of $x$ and the spectrum of $y$. In particular, one has $f(aa^*)a=af(a^*a)$ for every function $f\in \mathscr{C}$ (Sp$'(a^*a)$).

Let S = Sp($x$)$\cup$ Sp($y$). Proposition 6 of I, p. 106 implies that $x^*a=ay^*$. Consequently, one has $f(x)a=af(y)$ for every function $f$ of the form $z\mapsto P(z, z)$, where $P\in \mathbf{C}[X,Y]$ is a polynomial. Since the set of functions $f\in \mathscr{C}(S)$ satisfying $f(x)a=$ $af(y)$ is a closed subalgebra of $\mathscr{C}(S)$, it coincides with $\mathscr{C}(S)$ by TG, X, p. 40, Cor. 1.

The second assertion is a consequence of the first, applied to the hermitian elements $x=aa^*$ and $y=a^*a$, having regard to the fact that Sp$'(a^*a) =$ Sp$'(aa^*)$ (Prop. 1 of I, p. 5).