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
2 - Preliminaries
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 shall introduce all the basic materials and preliminary notions needed later on in this book.
C* and von Neumann algebras
For the details on the material of this section, the reader may be referred to [125], [40] and [76].
C*-algebras
An abstract normed *-algebra A is said to be a pre C*-algebra if it satisfies the C*-property : ‖x*x‖ = ‖x‖2. If A is furthermore complete under the norm topology, one says that A is a C*-algebra. The famous structure theorem due to Gelfand, Naimark and Segal (GNS) asserts that every abstract C*-algebra can be embedded as a norm-closed *-subalgebra of B(H) (the set of all bounded linear operators on some Hilbert space H). In view of this, we shall fix a complex Hilbert space H and consider a concrete C*-algebra A inside B(H). The algebra A is said to be unital or nonunital depending on whether it has an identity or not. However, even any nonunital C*-algebra always has a net (sequence in case the algebra is separable in the norm topology) of approximate identity, that is, an nondecreasing net eμ of positive elements such that eμa → a for all a ∈ A. Note that the set of compact operators on an infinite dimensional Hilbert space H, to be denoted by K(H), is an example of nonunital C*-algebra.
We now briefly discuss some of the important aspects of C*-algebra theory. First of all, let us mention the following remarkable characterization of commutative C*-algebras.
- Type
- Chapter
- Information
- Publisher: Cambridge University PressPrint publication year: 2007