norm of the logarithm of a positive element in $C^*$-algebra

Question

Suppose $a$ is an positive invertibel elment in a unital $C^*$-algebra.Why is the norm of $log(a)$ equal to max$\{log\|c\|,log\|c^{-1}\|\}$.The norm of $log(a)$ shoud be equal to spectral radius $r(log(a))$ of $log(a)$,but how to show that $r(log(a))=max\{log\|c\|,log\|c^{-1}\|\}$?

Answer 1

The spectrum of $\log a$ is $$ \sigma(\log a)=\{\log\lambda:\ \lambda\in\sigma(a)\}. $$ So, as the norm of a selfadjoint element is the maximum of the absolute value of the elements of the spectrum, $$ \|\log a\|=\max\{|\log\lambda|:\ \lambda\in\sigma(a)\}. $$ When $0<\lambda<1$, you have $$ |\log\lambda|=-\log\lambda=\log\lambda^{-1}. $$ Thus \begin{align} \|\log a\|&=\max\left\{ \{\log\lambda:\ \lambda\in\sigma(a)\cap[1,\infty)\} \cup \{\log\lambda^{-1}:\ \lambda\in\sigma(a)\cap(0,1)\}\right\}\\ &=\max\left\{ \{\log\lambda:\ \lambda\in\sigma(a)\cap[1,\infty)\} \cup \{\log\lambda:\ \lambda\in\sigma(a^{-1})\cap(1,\infty)\}\right\}\\ &=\max\left\{ \{\log\lambda:\ \lambda\in\sigma(a)\} \cup \{\log\lambda:\ \lambda\in\sigma(a^{-1})\}\right\}\\ &=\max\{\log\|a\|,\log\|a^{-1}\|\}. \end{align}