Spectral morphisms, K-theory, and stable ranks
We study the interplay between spectral morphisms, K-theory, and stable ranks for Banach algebras and, more generally, for Frechet algebras with open group of invertibles. After some preliminaries on the Gelfand transform, smooth subalgebras, and the property of Rapid Decay, we investigate closely four notions of ``noncommutative dimensions': the Bass, topological, connected, and general stable ranks. We show, among other things, that spectral morphisms preserve the latter two ranks. This answers the so-called Swan's problem. Stable ranks are relevant for stabilization phenomena in K-theory. We give new results in this direction. It is known that a spectrum-preserving morphism with dense image induces an isomorphism in K-theory. We introduce a weakening of the spectral property for morphisms, requiring that the preservation of the spectrum only holds over a dense subalgebra. We investigate such relatively spectral morphisms. We show that a relatively spectral morphism with dense image induces an isomorphism in K-theory. Furthermore, we generalize the usual K-functors - $K_0$ and $K_1$ - by defining certain spectral K-functors, and we prove that spectral K-groups are preserved by relatively spectral morphisms with dense image.