Let $A$ be a positive definite matrix such that $\|A\|>1$. Can we say that $A-I$ also is a positive matrix?
We can generalize this question for a unital $C^*$-algebra: Let ${\cal A}$ be a unital $C^*$-algebra with unit $1_{\cal A}$. If for some $a\in {\cal A}^+$ such taht $\|a\|>1$, can we say that $a -1_{\cal A}$ is positive?