Let $K$ be a complete nonarchimedean field. The classical Tate algebra $K\langle T \rangle$ has lots of automorphisms, e.g., any substitution $T\mapsto a\_1T+a\_2T^2+\cdots$, where $a\_1\in \mathcal{O}\_K^\times$, $a\_n\in\mathfrak{m}\_K$ for $n\geq 2$, and $a\_n\to 0$ as $n\to \infty$. 


This question is about perfectoid spaces. Let $K$ be a perfectoid field (in particular $K$ is a non-discrete valued field). Let $A$ be the ``perfectoid Tate algebra'' $K\langle T^{1/p^\infty}\rangle$. This is obtained by inverting $p$ on the $p$-adic completion of $\mathcal{O}\_K[T^{1/p^\infty}]=\bigcup\_{n\geq 1} \mathcal{O}\_K[T^{1/p^n}]$. 


If the characteristic of $K$ is $p$, then $K$ is perfect. In that case, the automorphisms of $K\langle T\rangle$ from the first paragraph extend uniquely to $A$. Other automorphisms arise by composing these with $p$th powers and roots. 


My question is, are there any other automorphisms? For instance, suppose $K$ has characteristic $p$ and $\varpi\in K$ has positive valuation. Then the substitution
$$ T\mapsto T+\varpi T^{1/p}+\varpi^2 T^{1/p^2}+\cdots $$
is certainly an endomorphism of $A$, but I cannot decide if it is an automorphism.



What are the automorphisms of a perfectoid Tate algebra?
Recall the following elementary fact in commutative algebra:


**Fact**: Say $f:R \to S$ is a map of commutative rings. Assume there exists a $p \in R$ that is a non-zero divisor in both $R$ and $S$, and that $R$ and $S$ are $p$-adically complete. Then $f$ is an isomorphism if and only if $f\_1$ is so; here $f\_n$ is reduction modulo $p^n$ of $f$.


[ *Proof*: Indeed, it is elementary, using the torsion freeness, to go from $f\_1$ being an isomorphism, to $f\_n$ being an isomorphism for all $n$, and then one takes a limit. ]


This immediately implies that the map mentioned in the question is an isomorphism (as it is defined on the integral perfectoid Tate algebra, and coincides with the identity modulo $\varpi$). In particular, there indeed are more automorphisms than the ones coming from Frobenius or the classical Tate algebra.


