Let A be a unital not-necessarily commutative algebra, defined over <mi mathvariant="doubl

kixEffinsoj

kixEffinsoj

Answered question

2022-06-20

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

Answer & Explanation

Lilliana Burton

Lilliana Burton

Beginner2022-06-21Added 19 answers

If α is an algebra automorphism, even one that is not assumed to take identity to identity, the multiplicative property of the map necessarily makes α ( 1 ) the identity of the image (which is equal to A) and so α ( 1 ) = 1.
So there are no algebra automorphisms that are strictly non-unital in the sense that they move the multiplicative identity. 1 α ( 1 ) = 0 for every algebra automorphism, and it is always non-invertible.

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