Book contents
- Frontmatter
- Contents
- Preface
- Notation
- 1 Introduction
- 2 Preliminaries
- 3 Quantum dynamical semigroups
- 4 Hilbert modules
- 5 Quantum stochastic calculus with bounded coefficients
- 6 Dilation of quantum dynamical semigroups with bounded generator
- 7 Quantum stochastic calculus with unbounded coefficients
- 8 Dilation of quantum dynamical semigroups with unbounded generator
- 9 Noncommutative geometry and quantum stochastic processes
- References
- Index
4 - Hilbert modules
Published online by Cambridge University Press: 19 January 2010
- Frontmatter
- Contents
- Preface
- Notation
- 1 Introduction
- 2 Preliminaries
- 3 Quantum dynamical semigroups
- 4 Hilbert modules
- 5 Quantum stochastic calculus with bounded coefficients
- 6 Dilation of quantum dynamical semigroups with bounded generator
- 7 Quantum stochastic calculus with unbounded coefficients
- 8 Dilation of quantum dynamical semigroups with unbounded generator
- 9 Noncommutative geometry and quantum stochastic processes
- References
- Index
Summary
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.
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- Quantum Stochastic Processes and Noncommutative Geometry , pp. 79 - 102Publisher: Cambridge University PressPrint publication year: 2007