Kuiper’s theorem for Hilbert modules. (English) Zbl 0616.46066
Operator algebras and mathematical physics, Proc. Summer Conf., Iowa City/Iowa 1985, Contemp. Math. 62, 429-435 (1987).
[For the entire collection see Zbl 0602.00004.]
We prove Kuiper’s theorem in full generality for Hilbert modules over arbitrary \(C^*\)-algebras: For any \(C^*\)-algebra A, the unitary group of the multiplier algebra \({\mathcal M}(K\otimes A)\) is contractible.
The proof is based on an analysis of projections in \({\mathcal M}(K\otimes A)\).
We prove Kuiper’s theorem in full generality for Hilbert modules over arbitrary \(C^*\)-algebras: For any \(C^*\)-algebra A, the unitary group of the multiplier algebra \({\mathcal M}(K\otimes A)\) is contractible.
The proof is based on an analysis of projections in \({\mathcal M}(K\otimes A)\).
MSC:
| 46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
| 46H25 | Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) |
| 46M10 | Projective and injective objects in functional analysis |
