On Tate Acyclicity and Uniformity of Berkovich Spectra and Adic Spectra

We construct an example of a uniform Banach algebra such that a rational localisation of the Berkovich spectrum does not preserve the uniformity. It is an example of a non-sheafy uniform Banach algebra. We also construct examples of uniform affinoid rings 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.