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Let $A$ be a unital Banach algebra. If $a$ is an element of $A$ with $||1-a||_{sp}<1$, then there exists $b\in A$ such that $b^2=a$. Furthermore, if $A$ is an involutive Banach algebra and if $a$ is hermitian, then a self-adjoint element can be chosen as the above $b$.

By above lemma, I have several question:

1- Would you give me a proof for it?

2-Would you give an example of a Banach algebra which is not involutive Banach algebra?

3-I know the fact that if an element $a\in A$ is in open unit ball $||1-a||<1$, then $a$ is invertable. By above lemma and this fact , I can conclude that every invertable element in a Banach algebra is positive. Am I right?

4- In an involutive Banach algebra, every invertable and self-adjoint element is positive. Is it correct?

Thanks for your attention.

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Conclusions drawn in $3,4$ are wrong. An example in $M_2(\mathbb{C})$:

$A=\begin{pmatrix} 1 &0\\ 0 & -1 \end{pmatrix}$

Then $A$ is invertible, self adjoint, but not positive.

To do $1$ note that by the given condition spectrum of $a$ is contained in $(0,1)$, and then apply functional calculus with the square root function.

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