We construct a non-sheafy uniform Banach algebra such that a rational localisation of the Berkovich spectrum does not preserve the uniformity. We also construct uniform affinoid rings in the sense of Roland Huber such that rational localisations of the adic spectra do not preserve the uniformity. One of them is an example of a non-sheafy uniform affinoid ring. We introduce the notion of local uniformity instead, and prove that the local uniformity implies the sheaf condition.

We establish class field theory for three-dimensional manifolds and knots. For this purpose, we formulate analogues of the multiplicative group, the idèle class group, and ray class groups in a cocycle-theoretic way. Following the arguments in abstract class field theory, we construct reciprocity maps and verify the existence theorems.

We study homotopy epimorphisms and covers formulated in terms of derived Tate’s acyclicity for commutative $C^*$-algebras and algebras of continuous functions valued in non-Archimedean valued fields. We prove that a homotopy epimorphism between commutative $C^*$-algebras precisely corresponds to a closed immersion between the compact Hausdorff topological spaces associated with them and a cover of a commutative $C^*$-algebra precisely corresponds to a topological cover of the compact Hausdorff topological space associated with it by closed immersions admitting a finite subcover. This permits us to prove derived and non-derived descent for Banach modules over commutative $C^*$-algebras.

For a complete valuation field k and a topological space X, we prove the universality of the underlying topological space of the Berkovich spectrum of the Banach k-algebra C bd (X, k) of bounded continuous k-valued functions on X. This result yields three applications: a partial solution to an analogue of Kaplansky conjecture for the automatic continuity problem over a local field, comparison of two ground field extensions of C bd (X, k), and non-Archimedean Gel'fand theory.

We give a criterion of the semisimplicity of a p-adic unitary representation of a topological monoid by the reduction of the associated operator algebra.