Let $A$ be a unital C*-algebra and $\{b_{n}\}$ be a positive invertible sequence in $A$. If $||1_{A}-b_{n}||\rightarrow 0$, can we conclude $||1_{A}-b_{n}^{-\frac{1}{2}}||\rightarrow 0$ ?
2026-03-28 10:15:55.1774692955
An exercise of positive element in C*-algebra
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Since $b_n\geq0$ and $b_n\to1$, eventually we will have $\sigma(b_n)\subset (k_1,k_2)$ for $k_1,k_2> 0$ and sufficiently close to $1$. Then $$ \|1_A-b_n^{-1/2}\|=\|b_n^{-1}(1_A-b_n)(1_A+b_n^{-1/2})^{-1}\|\\ \leq\|b_n^{-1}\|\,\|1_A-b_n\|\,\|(1_A+b_n^{-1/2})^{-1}\| \leq k_1^{-1}\,(1+k_2^{-1/2})^{-1}\,\|1_A-b_n\|$$