A question about $C^\ast$-algebra

Question

In Kadinson's book Fundamentals of The Theory of Operator Algebra, when the author proved the Theorem 7.2.1, he let $V$ be an extreme point of the unit ball of $C^\ast$-algebra $\cal{U}$, $h$ be a continuous function small on a small neighborhood of $t\in(0,1)$, vanishing outside that neiborhood, and non-zero at $t$. Then he said that

$$\|V^\ast V(I\pm h(V^\ast V))\|\leq1.$$

I do not understand why? Thanks to anyone who can give me a hint.

Answer 1

Note that $V^*V$ is a positive element with $\|V^*V\|=1$. This means that $\sigma(V^*V)\subset[0,1]$. If you consider the function $f(x)=x(1+h(x))$. If $h$ is chosen appropriately, then $|f(x)|\leq1$ for all $x$ (because the term $1+h(x)$ is positive only on the small neighbourhood of $t$).

Now functional calculus gives you $\sigma(V^*V(I+h(V^*V))\subset[0,1]$, which implies your inequality.

The case when subtraction is used instead of addition is exactly similar.