Banach modules, almost mathematics and condensed mathematics

Abstract: We study the relationship between almost mathematics, condensed mathematics and the categories of seminormed and Banach modules over a Banach ring $A$, with submetric (norm-decreasing) $A$-module homomorphisms for morphisms. If $A$ is a Banach ring with a norm-multiplicative topologically nilpotent unit $\varpi$ contained in the closed unit ball $A_{\leq1}$ such that $\varpi$ admits a compatible system of $p$-power roots $\varpi^{1/p^{n}}$ with \begin{equation*}\lVert\varpi^{1/p^{n}}\rVert=\lVert\varpi\rVert^{1/p^{n}}\end{equation*}for all $n$, we prove that the "almost closed unit ball" functor \begin{equation*}M\mapsto M_{\leq1}^{a}\end{equation*}is an equivalence between the category of Banach $A$-modules and submetric $A$-module maps and the category of $\varpi$-adically complete, $\varpi$-torsion-free almost $(A_{\leq1}, (\varpi^{1/p^{\infty}}))$-modules. We also obtain an analogous result for Banach algebras and almost algebras. The main novelty in our approach is that we show that the norm on the Banach module $M$ is completely determined by the corresponding almost $A_{\leq1}$-module $M_{\leq1}^{a}$, rather than being determined only up to equivalence. We deduce from our results the existence of a natural fully faithful embedding of the category of Banach $A$-modules and submetric $A$-module maps into the category of (static) condensed almost $(A_{\leq1}, (\varpi^{1/p^{\infty}}))$-modules in the sense of Mann, which factors through the full subcategory of solid condensed $(A_{\leq1}, (\varpi^{1/p^{\infty}}))$-almost modules. If $A$ is perfectoid and the adic spectrum of $(A, A^{\circ})$ is totally disconnected, we show that this embedding transforms the complete tensor product of Banach $A$-modules into (an almost analog of) the solid tensor product of solid condensed almost modules.