Functionals on \(\ell_2(A)\), Kuiper and Dixmier-Douady type theorems for \(C^*\)-Hilbert modules (Q2736088)

Let \(\ell_2(A)\) be the standard Hilbert \(C^*\)-module over a \(C^*\)-algebra \(A.\) Let us denote through \({\mathcal L}_A(\ell_2(A))\) the Banach algebra of all bounded A-homomorphisms of \(\ell_2(A),\) and through \({\mathcal L}_A^*(\ell_2(A))\) the \(C^*\)-algebra of operators admitting adjoint. Let \(GL(A)\) and \(GL^*(A)\) denote the correspondent groups of invertible operators. J. Cuntz and N. Higson proved the contractibility of \({\mathcal L}_A^*(\ell_2(A))\) for \(\sigma\)-unital \(C^*\)-algebra \(A.\) In the paper a simple proof of Cuntz-Higson theorem, distinguished from the original and based on a constraction of Newbauer's homotopy is given. This approach made possible to prove the contractibility of \({\mathcal L}_A(\ell_2(A))\) in some special cases, in particular, for \(A\) being a subalgebra of algebra of all compact operators in separable Hilbert space and for \(A=C(M),\) where \(M\) is finite-dimensional manifold, and to generalize the classical Dixmier-Douady theorem to the case \(GL(A)\) and \(GL^*(A)\) for \(\sigma\)-unital \(C^*\)-algebra \(A\) and the strict topology.NEWLINENEWLINENEWLINEReviewer's remark: The reviewed paper is the base of Ch.7 of the remarkable monograph [\textit{V. M. Manuilov} and \textit{E. V. Troitskij}, ``\(C^*\)-Hilbert modules''. M.:Factorial Press, 224 p. (Russian) (2001)].NEWLINENEWLINEFor the entire collection see [Zbl 0967.00102].