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Let $A$ be a unital not-necessarily commutative algebra, defined over $\mathbb{R}$ or $\mathbb{C}$. Take some $\alpha$ a non-unital algebra automorphism of $A$. Is it possible to find an example for $A$ and $\alpha$ such that $1-\alpha(1)$ is non-invertible in $A$?

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If $\alpha$ is an algebra automorphism, even one that is not assumed to take identity to identity, the multiplicative property of the map necessarily makes $\alpha(1)$ the identity of the image (which is equal to $A$) and so $\alpha(1)=1$.

So there are no algebra automorphisms that are strictly non-unital in the sense that they move the multiplicative identity. $1-\alpha(1)=0$ for every algebra automorphism, and it is always non-invertible.

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