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
3 - Quantum dynamical semigroups
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
Let us now restrict ourselves to the case when the general locally convex space X is replaced by a C* or a von Neumann algebra A, and study the implications of the complete positivity of a semigroup Tt acting on it.
Definition 3.0.1 A quantum dynamical semigroup (Q.D.S) on a C*-algebra A is a contractive semigroup Tt of class C0 such that each Tt is a completely positive map from A to itself. Tt is said to be conservative if Tt (1) = 1 for all t ≥ 0.
Generators of uniformly continuous quantum dynamical semigroups: the theorems of Lindblad and Christensen–Evans
For a uniformly continuous semigroup on a von Neumann algebra A ⊆ B(h), we have the following result.
Lemma 3.1.1Let Tt = etL be a uniformly continuous contractive semigroup acting on A with L as the generator. Then Tt is normal for each t if and only if L is ultra-strongly (and hence ultra-weakly) continuous on any norm-bounded subset of A.
Proof:
Let us first note that L is norm-bounded. If L is ultra-strongly continuous on bounded sets, then clearly etL is ultra-strongly continuous on bounded sets for each t, and hence normal. For the converse, first note that for any t ≥ 0 and x ∈ A, we have
Hence it is not difficult to see that
Now suppose that xα is a net of elements in A such that xα strongly converges to x ∈ A and there exists positive constant M such that ∥xα∥ ≤ M for all α. Fix u ∈ h and ∈ > 0. Choose t0 small enough so that ∥L∥2M∥u∥t0 ≤ ∈.
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- Quantum Stochastic Processes and Noncommutative Geometry , pp. 33 - 78Publisher: Cambridge University PressPrint publication year: 2007