One difference between
Hilbert modules and Hilbert spaces is that Hilbert modules are not “self-dual” in
general. Another difference is that Hilbert modules are not orthogonally
complementary. Let H be a Hilbert module over a C∗-algebra A. We show that if A
is monotone complete then H∗, the “dual” of H, can be made into a self-dual Hilbert
A-module. We also show that if H is full and countably generated, then H is
orthogonally complementary if and only if every bounded module map in H has an
adjoint. It turns out that these results are closely related to the problem of extensions
of bounded module maps. Let C1 be the category whose objects are Hilbert
A-modules and morphisms are contractive module maps with adjoints, and
C2 the category whose objects are Hilbert A-modules and morphisms are
contractive module maps. We find that injective modules in the category
whose objects are Hilbert A-modules and morphisms are contractive module
maps. We find that injective modules in the category C1 are precisely those
that are orthogonally complementary. We show that Hilbert modules over a
monotone complete C∗-algebra are injective in C2 if and only if they are
self-dual. We also show that if A is not an AW∗-algebra then A itself is
not injective A-module in the category C2. A few related results are also
included.
