Explicit concrete examples of k-affinoid algebras I am having some troubles understanding $k$-affinoid algebras (where ($k$, |.|) is a complete, non Archimedean field, |.| is not trivial) and i am looking for some more concrete and particular examples for better understanding. I was wondering if we can find some $k$-affinoid algebras $(A,||.||)$ where : 


*

*||.|| is not power multiplicative

*||.|| is Archimedean.

*$A$ is reduced and ||.|| is not multiplicative.


An example or a construction would really help, thanks
 A: A bit old, but maybe this still helps someone.
I'll start with your second example: Find some $K$-affinoid algebra $A$ such that the norm is not non-archimedean.
For this consider $K\langle T \rangle = \{\sum_{n \in \mathbb N} a_nT^n \in K[[T]] \mid \sum_{n \in \mathbb N} |a_n| < \infty \}$, where the norm is given by $\|\sum_{n \in \mathbb N} a_nT^n\| = \sum_{n \in \mathbb N} |a_n|$. This is a Banach $K$-algebra. The quotient
$$A := K\langle T \rangle /(T^2)$$
equipped with the residue norm is a Banach $K$-algebra again, as $(T^2)$ is closed. Write $\mathbb T$ for the Tate-algebra $K\{T\}$ with the Gauss-norm. Then 
$$\mathbb T \to A, \sum_{n \in \mathbb N} a_nT^n \mapsto [a_1T + a_0]$$
is a well-defined and continuous, as $\|[T]\|_{res} = 1 = \|T\|_{Gauss}$, $K$-linear and obviously surjective. In particular it's admissible, which shows that $A$ is indeed a (strictly) $K$-affinoid algebra. However the residue norm on it is not non-archimedean, as for example
$$2 = \|[1 + T]\|_{res} > 1 = \max\{\|[1]\|_{res}, \|[T]\|_{res}\}.$$

For the first and third example it obiously suffices to find an example that is reduced and not power-multiplicative. I don't have a full proof, but $K\{2^{-1}T\}/(T^2-1)$ equipped with residue norm (obviously $K$-affinoid as $(T^2-1)$ is closed) should do the trick, and show that it's not power-multiplicative by showing that $\|[T^2]\|_{res} \neq \|[T]\|_{res}^2$.
