I'm studying "MURPHY, $C^*$-Algebras and Operator Theory" thoroughly and got stuck in the following exercise:
Exercise 2, chapter 2. Let $A$ be a unital $C^*$-algebra.
(a) If $a,b$ are positive elements of $A$, show that $\sigma(ab)\subseteq\mathbb{R}^+=\left\{x\in\mathbb{R}:x\geq 0\right\}$.
(b)If $a$ is an invertible element of $A$, show that $a=u|a|$ for a unique unitary $u$ of $A$. Give an example of an element of $B(H)$ for some Hilbert space $H$ that cannot be written as a product of a unitary times a positive operator.
(c)Show that if $a\in Inv(A)$, then $\Vert a\Vert=\Vert a^{-1}\Vert=1$ if and only if $a$ is a unitary.
Item (b) is easy: the only possibility for $u$ is $u=a(a^{-1})(a^*)^{-1}|a|=(a^*)^{-1}|a|$, which is readily verified to be unitary. For the example, let $a$ be any non-invertible isometry in a Hilbert space $H$ (e.g. $a$ is the unilateral shift in $l^2(\mathbb{Z}_+)$). If $a=up$, $u$ unitary and $p$ positive, then $1=a^*a=p^*u^*up=p^2$, so $p=(1)^{1/2}=1$, hence $a=u$, absurd.
Now, I really have no idea how to proceed with items (a) and (c). If $A$ were abelian, both would be trivial (using the Gelfand representation). I believe that in (c) we have to use the representation given in (b). Actually, to use the Gelfand representation in (c), it suffices to show that $a$ is normal, but I don't see how the hipothesis on the norms would imply that.
For (a), you have to use that $\sigma(xy)\cup\{0\}=\sigma(yx)\cup\{0\}$ for any two operators $x,y$. Then $$ \sigma(ab)=\sigma((ab^{1/2})b^{1/2})\subset\sigma(b^{1/2}ab^{1/2})\cup\{0\}\subset\mathbb R^+. $$ The point is that $b^{1/2}ab^{1/2}$ is positive.
For (c), \begin{align} \max\sigma(a^*a)&=\|a^*a\|=\|a\|^2=1=\|a^{-1}\|^2\\ &=\|(a^{-1})^*a^{-1}\|=\|(aa^*)^{-1}\|=1/\min\sigma(aa^*)=1/\min\sigma (a^*a). \end{align} Thus $a^*a $ is positive with spectrum $\{1\} $, so $a^*a=1$. Similarly, $aa^*=1$.