strong convergence in Hilbert c* module

Question

For a monotone sequence of projections in Hilbert c* module, do we have similar conclusion as in Hilbert space (such as strong convergence)? If not, what could be said in that situation?

Answer 1

I'm not sure what you mean by a projection "in" a Hilbert module --- an element of the module that is a projection, or a projection of the module onto a submodule? The first option has no meaning in general. In any case, remember that any unital C*-algebra is a left Hilbert module over itself, and can also be identified (via left multiplication maps) as the algebra of bounded left module operators. So nothing more can be true in general than is true for arbitrary unital C*-algebras. In particular, an increasing sequence of projections need not converge in any sense.

Maybe it is helpful to add that if $E$ is a Hilbert module over a C*-algebra $A$ and $\phi$ is a state on $A$, then $[x,y] = \phi(\langle x,y\rangle_A)$ defines a pre-inner product on $E$. So after factoring out null vectors and completing it becomes a Hilbert space. This can be a way to approach general Hilbert modules using ideas about ordinary Hilbert spaces.

Another comment is that self-dual Hilbert modules over von Neumann algebras behave much more like Hilbert spaces in these kinds of ways ... a good reference is still Paschke's original paper, "Inner product modules over B*-algebras", Trans. Amer. Math. Soc. 182 (1973), 443-468.