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

TL;DR

This paper introduces an 'almost' version of Auslander regularity and applies it to establish the regularity of key $p$-adic Banach algebras relevant in $p$-adic representation theory, including Weyl and distribution algebras.

Contribution

It develops an 'almost' regularity concept and proves Auslander regularity for several important $p$-adic Banach algebras used in representation theory.

Findings

Proves Auslander regularity for completed Weyl algebras

Establishes regularity for completed enveloping algebras of Lie algebras

Shows regularity of the Banach completion of distribution algebras

Abstract

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 $G$.