ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 18, 46, 47 Helemskii, A. Ya., Moscow State University, Russia: Description of projective Hilbert modules over C^{∗}-algebras A Hilbert module over a C^{∗}-algebra A is a Hilbert space with an involutive representation of A. We present a complete description of projective (as Banach A-modules) Hilbert A-modules. In particular, such a module is isomorphic to a Hilbert sum of principal ideals in A, generated by minimal projections, with a finite number of summands, isomorphic to the same infinite-dimensional module. As a corollary, we describe all spatially projective operator C^{∗}-algebras on H (i. e. those with projective H). In particular, an irreducible A is spatially projective iff A ⊇ K(H) . The standard form of B(H); dim H = ∞ is not spatially projective. Applications to the cohomology of C^{∗}-algebras, related to the similarity representation problem, will be discussed.