3 - Quantum dynamical semigroups

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 T_t acting on it.

Definition 3.0.1 A quantum dynamical semigroup (Q.D.S) on a C^*-algebra A is a contractive semigroup T_t of class C₀ such that each T_t is a completely positive map from A to itself. T_t is said to be conservative if T_t (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 T_t = e^tL be a uniformly continuous contractive semigroup acting on A with L as the generator. Then T_t 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 e^tL 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 x_α is a net of elements in A such that x_α strongly converges to x ∈ A and there exists positive constant M such that ∥x_α∥ ≤ M for all α. Fix u ∈ h and ∈ > 0. Choose t₀ small enough so that ∥L∥²M∥u∥t₀ ≤ ∈.