How does an analytic space correspond to a $p$-adic Banach space

Question

Let $K$ be a finite extension of $\mathbb{Q}_p$, and $V$ be a Banach algebra over $K$, then what is the $K$-analytic space corresponding to $V$? What is the definition of $K$-analytic space? This is mentioned in the first paragraph on page 6 in Laurent Berger's paper.

Similarly, if $W$ is a Banach algebra over $\mathbb{C}$, then does $W$ correspond to a complex analytic space? I only know for a variety over $\mathbb{C}$, we can do this and we have GAGA.

Thanks!

Answer 1

Without reading much beyond the pages adjacent to page 6, but looking at [ST3] (especially seeing the references [BGR] and [NFA]), I believe Berger-Colmez are using $K_n$-analytic spaces in the sense of Berkovich. Without rehashing the details here, the idea is to extend the ideas introduced by Tate's rigid analytic geometry to make for a theory that is more flexible, allowing for proper construction of affinoid charts, which are the Banach-analytic analog of affine schemes. These may be assembled into atlases (in the sense of differential geometry) for actual topological spaces. This contrasts with the Grothendieck topologies on Tate's (strictly) affinoid algebras.

See:

1. Berkovich, Spectral Theory and Analytic Geometry over Non-Archimedean Fields. Here, he introduces the $K_n$-analytic spaces, building on BGR and Tate (among others). I found this to be a difficult starting reference, however, and recommend:
2. B. Conrad, Several approaches to non-Archimedean gometry.

Just as a side note regarding your question about $\mathbb C$-analytic spaces: the closest analogy for Banach algebras over $\mathbb C$ with Archimedean norms are the maximal ideal spaces (and more generally, primitive ideal spaces) of a Banach algebra. These come up prominently in abstract harmonic analysis. See: Rudin, Fourier analysis on groups.