  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.
