Relatively spectral morphisms and applications to K-theory

TL;DR

This paper studies relatively spectral morphisms between Banach algebras, establishing a relative Density Theorem for K-theory isomorphisms and solving Swan's problem for higher connected stable ranks.

Contribution

It introduces the concept of relatively spectral morphisms, proves a relative Density Theorem, and extends Swan's problem to a hierarchy of higher connected stable ranks.

Findings

Proves a relative version of the Density Theorem for K-theory.

Solves Swan's problem for the hierarchy of higher connected stable ranks.

Establishes conditions under which spectral information over dense subalgebras suffices for K-theory isomorphism.

Abstract

Spectral morphisms between Banach algebras are useful for comparing their K-theory and their "noncommutative dimensions" as expressed by various notions of stable ranks. In practice, one often encounters situations where the spectral information is only known over a dense subalgebra. We investigate such relatively spectral morphisms. We prove a relative version of the Density Theorem regarding isomorphism in K-theory. We also solve Swan's problem for the connected stable rank, in fact for an entire hierarchy of higher connected stable ranks that we introduce.