Hilbert C *‐Modules over Monotone Complete C *‐Algebras

The aim of the present paper is to describe self‐duality and C *‐reflexivity of Hilbert A‐modules ℳ over monotone complete C *‐algebras A by the completeness of the unit ball of ℳ with respect to two types of convergence being defined, and by a structural criterion. The derived results generalize earlier results of H. Widom [Duke Math. J. 23, 309‐324, MR 17 # 1228] and W. L. Paschke [Trans. Amer. Mat. 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. M. Ozawa [J. Math. Soc. Japan 36, 589‐609, MR 85 # 46068]). Especially, one derives that for a C *‐algebra A the A‐valued inner product of every Hilbert A‐module ℳ can be continued to an A‐valued inner product on it's A‐dual Banach A‐module ℳ' turning ℳ' to a self‐dual Hilbert A‐module if and only if A is monotone complete (or, equivalently, additively complete) generalizing a result of M. Hamana [Internat. J. Math. 3 (1992), 185 ‐ 204]. A classification of countably generated self‐dual Hilbert A‐modules over monotone complete C *‐algebras A is established. The set of all bounded module operators End ′(ℳ) on self‐dual Hilbert A‐modules ℳ over monotone complete C *‐algebras A is proved again to be a monotone complete C *‐algebra. Applying these results a Weyl‐Berg type theorem is proved.