Stable ranks, 𝐾-groups and Witt groups of some Banach and 𝐶*-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 π 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

Spectral morphisms, K-theory, and stable ranks

“…A variant of the Corach -Larotonda fibration, where the base space Lg n (A) is replaced by the subspace Lc n (A) consisting of last columns of invertible n-by-n matrices, is used by Rieffel in [32] and later on by Thomsen [40] and Badea [4], among others. This allows for the following sharpening of the injectivity bound from Theorem 6.4: the map [4]). A downside of this improvement is that the general stable rank is, in general, more difficult to compute than the connected stable rank.…”