I'm working through some exercises from Ch.2 of C* algebras By Murphy, and I'm stuck on the very first part of a problem. The problem is the following:
Let $A$ be a unital $C^{*}$ algebra. If $r(a) < 1$, and $b = \left(\sum_{n=0}^{\infty} a^{{*n}} a^n \right)^{\frac{1}{2}}$, show that $b \geq I$, and that $||bab^{-1}|| < 1$.
So I don't really wan't help on the exercise, but I do want to understand how the setup of the problem even makes sense.
First off, what if $a=0$? Then $r(a) = 0 < 1$, but then $b=0$, and hence $b \ngeq I$ and $b^{-1}$ doesn't exist, so the problem statement seems nonsensical.
Secondly, I don't see how $r(a) < 1$ garentees the existence of $b \in A$. Since we're in a Banach space, it suffices to verify that $\left| \left|\sum_{n=0}^{\infty} a^{*n}a^n \right| \right| < \infty$, and then invoke the existence of the square root of the element $\sum_{n=0}^{\infty} a^{*n}a^n$. However, doing this I yield: $$\left| \left|\sum_{n=0}^{\infty} a^{*n}a^n \right| \right| \leq \sum_{n=0}^{\infty} ||a^{*n}a^n|| = \sum_{n=0}^{\infty} ||a^n||^2 \leq \sum_{n=0}^{\infty} ||a||^{2n}.$$ However, for the righthand sum to converge, I need that $||a|| <1$, and $r(a) < 1$ doesn't guarantee this, as $a$ isn't assumed normal (so that $||a|| = r(a)$).
Is tthe problem missing some assumptions that aren't stated, or am I missing something obvious?
Thank you.
For your first question, $a^0=I$. So when $a=0$, $b=I$, and in general $b\geq I$.
As for your second questions, the key is that $$ r(a)=\lim_n\|a^n\|^{1/n}=\lim_n\|a^{*n}a^n\|^{1/2n}. $$ You should be able to conclude from this that $\|a^{*n}a^n\|<1$ for $n$ big enough.