Stable Ranks, K-Groups and Witt Groups of some Banach and C-star Algebras

Abstract

We show that certain dense and spectral invariant subalgebras of a $C^*$-algebra have the same bilateral Bass stable rank. This is a partial answer for (a version of) an open problem raised by R.G. Swan. Then, for certain Banach algebras, we indicate when the homotopy groups $\pi_{i}(GL_{n}(A))$ stabilize for large $n$. This is an improvement of a result due to G. Corach and A. Larotonda. Using some results due to M. Karoubi, we show the isomorphism of the Witt group of a symmetric Banach algebra with the $K_0$-group of its enveloping $C^*$-algebra. The question if this is true for all involutive Banach algebras was raised by A. Connes.