Hilbert C*-modules over monotone complete C*-algebras

Abstract

The aim of the present paper is to describe self-duality and C*- reflexivity of Hilbert {\bf A}-modules $\cal M$ over monotone complete C*-algebras {\bf A} by the completeness of the unit ball of $\cal M$ with respect to two types of convergence being defined, and by a structural criterion. The derived results generalize earlier results of {\sc H.~Widom} [Duke Math.~J.~23, 309-324, MR 17 \# 1228] and {\sc W.~L.~Paschke} [Trans. Amer.~Math.~Soc.~182, 443-468, MR 50 \# 8087, Canadian J.~Math.~26, 1272-1280, MR 57 \# 10433]. For Hilbert C*-modules over commutative AW*-algebras the equivalence of the self-duality property and of the Kaplansky-Hilbert property is reproved, (cf. {\sc M.~Ozawa} [J.~Math.~Soc.~Japan 36, 589-609, MR 85m:46068] ). Especially, one derives that for a C*-algebra {\bf A} the {\bf A}-valued inner pro\-duct of every Hilbert {\bf A}-module $\cal M$ can be continued to an {\bf A}-valued inner product on it's {\bf A}-dual Banach {\bf A}-module $\cal M$' turning $\cal M$' to a self-dual Hilbert {\bf A}-module if and only if {\bf A} is monotone complete (or, equivalently, additively complete) generalizing a result of {\sc M.~Hamana} [Internat.~J.~Math.~3(1992), 185-204]. A classification of countably generated self-dual Hilbert {\bf A}-modules over monotone complete C*-algebras {\bf A} is established. The set of all bounded module operators ${\bf End}_A (\cal M)$ on self-dual Hilbert {\bf A}-modules $\cal M$ over monotone complete C*-algebras {\bf A} is proved again to be a monotone complete