Berkovich spectra of elements in Banach rings

Abstract

Extending the notion of the spectrum Σf of an element f in an ultrametric Banach algebra over a complete valued field (as defined by Berkovich), we introduce and briefly study the Berkovich spectrum σR,SBer(a) of an element a in a unital Banach algebra R over a commutative unital Banach ring S (in particular, when R is a Banach ring). This spectrum is a compact subset of the affine analytic line AS1 over S, and the latter can be identified with the suitable equivalence classes of all elements in all complete valued fields that are Banach S-algebras. If R is generated by a as a unital Banach S-algebra, then σR,SBer(a) coincides with the spectrum M(R) of R. If R is a unital complex Banach algebra but we regard it as a Banach ring only, then σR,Z1Ber(a) is the quotient of the ordinary spectrum σRC(a) by complex conjugation.For a non-Archimedean complete valued field k and an infinite dimensional ultrametric Banach k-space E with an orthogonal basis, if u∈L(E) is a completely continuous operator, we show that several different ways to define the spectrum of u give the same compact set σL(E),kBer(u). As an application, we give a relation between σL(E),kBer(u) and zeros of the Fredholm determinant det⁡(1−tu) (as defined by Serre) in complete valued field extensions of k.