Quantum Dynamical Systems from the Point of View of Noncommutative Mathematics 157
Such approximately invariant sets in an arbitrary group are called
Følner sets; their existence is equivalent to theamenabilityof the
group in question, that is, the existence of a state on the algebra
`∞(Γ)that is invariant under the left and right translations. In
particular, as mentioned before,Z is amenable. The argument
above is just the tip of an iceberg of connections between
amenability and finite-dimensional approximations in the theory
ofC∗-algebras. Several examples of such connections can be found
in [BrO]; in fact Theorem 2.4.2 can be generalized to the crossed
products by actions of arbitrary locally compact amenable groups
(Chapter 8 of [NS]) and even discrete amenablequantum groups
([SZ 3 ]).
Of course the simplest and in a sense the most interesting
example of the application of the above theorem is obtained by
puttingβ=α.
Corollary 2.4.3 LetAbe a nuclearC∗-algebra, letα∈Aut(A)and
let Adu∈Aut(AoαZ)be the canonical automorphism extending
α. Then htα=ht Adu.
The corollary means that computing the Voiculescu entropy of
inner automorphisms is as complicated as that of general
automorphisms (as each automorphism can be ‘made inner’ via
the crossed product extension and this procedure does not change
the Voiculescu entropy).
2.5 Quantum ‘Measurable’ Dynamical Systems and
Classical Ergodic Theorems
For the last two sections of these lectures, we switch our attention
from the entropy considerations and topological dynamics to
ergodic theorems and measurable dynamics.
2.5.1 Measurable dynamical systems and individual ergodic
theorem
So far, we talked mainly about topological dynamical systems.
Classically, an equally important class is that of so-called
measurable dynamical systems, that is, pairs((Y,μ),T), where(Y,μ)
is a probability space and T : (Y,μ) → (Y,μ) is a measure
preserving invertible transformation(this means that bothTandT−^1