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. Swanʹs problem (Have two Banach or good Fréchet algebras as in the density theorem inK-theory the same stable rank?) are obtained. For example, a Fréchet dense ∗-subalgebraAof aC*-algebraB, closed underC*-functional calculus of self-adjoint elements, has the same Bass stable rank asB.