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Let $A \subseteq B$ be two associative commutative $k$-algebras with $1$.

Let $f$ be an involution on $A$, namely, a $k$-algebra automorphism of $A$ of order two.

Can one tell when $f$ can be extended to an involution on $B$ (not just to an automorphism of $B$, but to an order two automorphism of $B$)? Is there a known criterion for extending $f$?

Special cases: What if we further assume that both $A$ and $B$ are unique factorization domains (UFD's)? Or both are fields?

An example I have in mind: $A=k[x_1,\ldots,x_n]$, $B=[[k[x_1,\ldots,x_n]]$ (= formal power series); I think that in this case it is possible to extend any involution on $A$ to an involution on $B$.

A somewhat related question is this question.

Moreover, what about the converse:

If $g$ is an involution on $B$, can one tell when it can be restricted to an involution on $A$? Notice that the restriction of $g$ to $A$ will be 'automatically' an involution on $A$, once it is shown that $g(A) \subseteq A$.

Any comments are welcome!

user237522
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    Does the involution $x\mapsto -x+a$, $0\neq a\in k$ of $k[x]$ extend to $k[[x]]$? – Mohan Jan 18 '18 at 22:39
  • If you 'ask', then I guess you hint that it can not be extended to $k[[x]]$.. – user237522 Jan 18 '18 at 22:43
  • $f: x \mapsto -x+a$, $0 \neq a \in k$, is not a $k$-endomorphism of $k[[x]]$: $-x+a$ is a unit of $k[[x]]$, $x$ is not a unit of $k[[x]]$ (https://math.stackexchange.com/questions/644468/ring-of-formal-power-series-over-a-field-is-a-principal-ideal-domain), and it is not possible that a unit $-x+a$ will be mapped to a non-unit $x$. – user237522 Jan 18 '18 at 23:29
  • Glad you realized that. So, may be you should ask a different question? – Mohan Jan 19 '18 at 01:19
  • Thank you! So the example I had in mind is not relevant for extensions (what about restricting involutions from $k[[x]]$ to $k[x]$?). I hope that there exist nice examples where it is possible to extend or restrict involutions (maybe in certain fields?). – user237522 Jan 19 '18 at 02:29
  • The following question is relevant: https://mathoverflow.net/questions/248234/involutions-on-power-series-mathbbcx-1-ldots-x-n and perhaps it shows that every involution on $\mathbb{C}[[x_1,\ldots,x_n]]$ can be restricted to an involution on $\mathbb{C}[x_1,\ldots,x_n]$ (if the conjugation will not cause any problems). – user237522 Jan 19 '18 at 02:36

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