ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 46, 43 Ylinen, Kari, University of Turku, Finland Positive definite operator functions on groups and related multilinear operators on products of Banach-^{∗}-algebras For k = 1, …, r, let A_{k} be a Banach^{∗}-algebra with an approximate identity, and let B be a C^{∗}-algebra. An r-linear map Φ : A_{1} × … × A_{r} → B is said to be Schur-completely positive if Φ_{n}(C_{1}^{∗}C_{1}, …, C_{r}^{∗}C_{r}) ≥ 0 for all positive integers n and all C_{k} ∈ M_{n}(A_{k}), k = 1, …, r, where Φ_{n}(C_{1}, …, C_{r}) = (Φ(c_{ij1}, …, c_{ijr})) for C_{k} = (c_{ijk}). There is a Stinespring type representation theorem for such maps, and if each A_{k} is commutative, the positivity of Φ already implies its Schur-complete positivity. For a Hilbert space H, positive definite L(A_{r}, L(H))-valued functions on locally compact groups in the sense of Holevo are closely related to Schur-completely positive maps, and the representation theorem for the latter can be used to prove a Riesz type decomposition theorem for such functions.