The stable rank of topological algebras and a problem of R. G. Swan

Various notions of stable ranks are studied for topological algebras. Some partial answers to R. G. Swan's problem (Have two Banach or good Frechet al gebras as in the density theorem in K-theory the same stable rank?) are obt ained. For example, a Frechet dense *-subalgebra A of a C*-algebra B, close d under C-infinity-functional calculus of self-adjoint elements, has the sa me Bass stable rank as B. (C) 1998 Academic Press.