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.
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- June 1999
- DOI:
- 10.48550/arXiv.math/9906073
- arXiv:
- arXiv:math/9906073
- Bibcode:
- 1999math......6073B
- Keywords:
-
- Mathematics - Operator Algebras;
- 46H05;
- 46H10
- E-Print:
- 19 pages, to appear in Contemporary Math