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The following question is a natural continuation of this MSE question; further elaboration can be found in it.

I have now edited my current question, according to the two nice comments that I have received:

Let $k$ be a field of characteristic zero.

Let $f: (x,y) \mapsto (p,q)$ be a $k$-algebra endomorphism of $k[x,y]$, $I=\langle p,q \rangle$ a proper ideal of $k[x,y]$ and $\delta$ an involution on $k[x,y]$, namely, a $k$-algebra automorphism of order two.

Assume that $\delta(I) \subseteq I$.

I do not mind to further assume that:

(i) $\delta: (x,y) \mapsto (x,-y)$ (ii) $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in k-\{0\}$.

Is the following claim true?

Claim: There exist automorphisms $g,h$ of $k[x,y]$ such that one of $\{(gfh)(x), (gfh)(y)\}$ is symmetric or skew-symmetric w.r.t. some involution on $k[x,y]$.

An attempt to answer my question:

Write $p=s_1+k_1$ and $q=s_2+k_2$, with $s_1,s_2$ symmetric and $k_1,k_2$ skew-symmetric.

$\delta(\langle p,q \rangle) \subseteq \langle p,q \rangle$, implies that $s_1,k_1,s_2,k_2 \in \langle p,q \rangle$.

Then we can write: (i) $s_1=Ap+Bq=(A_s+A_k)(s_1+k_1)+(B_s+B_k)(s_2+k_2)$, for some $A=A_s+A_k$, $B=B_s+B_k$, with $A_s,B_s$ symmetric, $A_k,B_k$ skew-symmetric.

By applying $\delta$ to (i) we get (i)', and then (i)+(i)' and (i)-(i)' yield: $s_1=A_s s_1+A_k k_1+B_s s_2+ B_k k_2$ and $0=A_s k_1+A_k s_1+B_s k_2+B_k s_2$.

Similarly for $k_1,s_2,k_2$.

However, I am not sure if this helps (hopefully yes).

Perhaps I am missing a relevnt known result that can answer my question?

What about $C$*-algebras? What about the first Weyl algebra? Notice that the subalgebra $k[p,q]$ is not invariant under $\delta$, unless $p,q$ are of one of the above three forms of $(gfh)(x),(gfh)(y)$.

The one-dimensional case:

Notice that in $k[x]$ an analog result holds for the involution $\delta_c: x \mapsto -x+c$, $c \in k$, namely:

If $I$ is an ideal of $k[x]$ which is invariant under $\delta_c$, then $I=\langle h \rangle$, with $h$ symmetric or skew-symmetric with respect to $\delta_c$.

The proof of this one-dimensional case can be found in the above mentioned MSE question.


This is a slightly similar question (with $\mathbb{F}_2[x,y]$ instead of $k[x,y]$, $p,q$ homogeneous, etc.).

Thank you very much!

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    $\begingroup$ 18 versions in two days. $\endgroup$ – Gerry Myerson Sep 12 at 22:10
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    $\begingroup$ mathoverflow.net/posts/341355/revisions shows 18. $\endgroup$ – Gerry Myerson Sep 12 at 22:15
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    $\begingroup$ Come off it. You were already up to version 13 well before the edits during that hour. $\endgroup$ – Gerry Myerson Sep 13 at 9:50
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    $\begingroup$ "Minor edits"? 13 added 108 characters in body; 12 added 37 characters in body; 8 deleted 143 characters in body; edited tags; 7 deleted 143 characters in body; edited tags; 6 added 189 characters in body; 5 added 48 characters in body; 4 deleted 74 characters in body; 2 deleted 45 characters in body. $\endgroup$ – Gerry Myerson Sep 14 at 23:17
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    $\begingroup$ You say $k$ is merely a field of characteristic zero, hence it could be the rationals or the p-adic numbers, neither of which will give rise to Cstar algebras. Do you actually know the definitions involved? In fact $A_1(k)$ cannot be a normed algebra, let alone a Cstar algebra. I fear that this contributes to an impression that you are trying to attack the Jacobian conjecture by finding things from various parts of maths to throw at the problem $\endgroup$ – Yemon Choi Sep 15 at 1:42

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