Auslander regularity of $p$-adic Banach algebras via almost mathematics

We discuss an “almost” version of Auslander regularity and use it to prove the Auslander regularity of various Banach algebras over non-discretely valued fields appearing naturally in $p$-adic locally analytic representation theory: completed Weyl algebras, the completed enveloping algebra of a Lie algebra, and the Banach completion of the distribution algebra $D(G,K)$ for a compact $p$-adic Lie group $G$.