Stable Ranks, KGroups and Witt Groups of some Banach and Cstar 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 1999
 arXiv:
 arXiv:math/9906073
 Bibcode:
 1999math......6073B
 Keywords:

 Mathematics  Operator Algebras;
 46H05;
 46H10
 EPrint:
 19 pages, to appear in Contemporary Math