In this chapter we introduce the idea of Hilbert modules and briefly discuss some useful results on them. For a more detailed account on this subject, the reader is referred to [81], [90], [98] (and [122] for von Neumann modules).
Hilbert C*-modules
A Hilbert space is a complex vector space equipped with a complex-valued inner product. A natural generalization of this is the concept of Hilbert module, which has become quite an important tool of analysis and mathematical physics in recent times.
Definition 4.1.1 Given a *-subalgebra A ⊆ B(h) (where h is a Hilbert space), a semi-Hilbert A-module E is a right A-module equipped with a sesquilinear map 〈., .〉 : E × E → A satisfying 〈x, y〉* = 〈y, x〉, 〈x, ya〉 = 〈x, y〉a and 〈x, x〉 ≥ 0 for x, y ∈ E and a ∈ A. A semi-Hilbert module E is called a pre-Hilbert module if 〈x, x〉 = 0 if and only if x = 0; and it is called a Hilber C*-module if furthermore A is a C*-algebra and E is complete in the norm x → ∥〈x, x〉∥1/2 where ‖.‖ the C*-norm of A.
It is clear that any semi-Hilbert A-module can be made into a Hilbert module in a canonical way: first quotienting it by the ideal {x : 〈x, x〉 = 0} and then completing the quotient.
Let us assume that A is a C*-algebra. The A-valued inner product 〈., .〉 of a Hilbert module shares some of the important properties of usual complexvalued inner product of a Hilbert space, such as the Cauchy–Schwartz inequality, which we prove now.