Quantum Dynamical Systems from the Point of View of Noncommutative Mathematics 153
the automorphismαcan be always viewed as an action ofZ. As
one can check (using some straightforward norm computations),
the construction above does not depend (up to an isomorphism) of
the choice of the initial representation ofAonH. We will always
use the notationι :A →AoαZto describe the embedding ofA
into the relevant crossed product and write u for the
corresponding unitary. Sometimes it will be more convenient to
identify the sequence (ξn)n∈Z with the (Hilbert space norm)
convergent series∑n∈Zδn⊗ξnand work directly with individual
vectorsδk⊗ξ∈`^2 (Z)⊗H(k∈Z,ξ∈H).
Exercise 2.4.1 Formalize the remark on the independence of the
construction ofAoαZof the choice of the original representation
ofA. Note that we can also speak ofι:A→AoαZandu∈AoαZ
without the risk of confusion.
Exercise 2.4.2 Can you identify theC∗-subalgebra of AoαZ
generated by the operatoru(see Section 2.1.5)? In particular, what
isCoidCZ?
Consider the following computation (a∈A,ξ∈H,k∈Z) :
(u∗ι(a)u)(δk⊗ξ) = (u∗ι(a))(δk− 1 ⊗ξ) =u∗(δk− 1 ⊗α^1 −k(a)ξ)
=δk⊗α^1 −k(a)ξ=δk⊗α−k(α(a))ξ
=ι(α(a))(δk⊗ξ).Thus, identifying A with ι(A), we observe that the inner
automorphism AduofAoαZ‘extends’α:
(Adu)◦ι=ι◦α. (2.4.1)It can be further shown that AoαZ is in fact the universal
C∗-algebra containing a copy ofAand a unitaryu‘implementing’
the actionα. This is related to the fact thatZisamenable– see the
comments after Theorem 2.4.2.
Remark 2.4.1 Equality (2.4.1) implies in particular that the
‘normally ordered’ set{ι(a)ul:a∈A,l∈Z}is linearly dense in
AoαZ.
Exercise 2.4.3 Let(α,A)be a noncommutative dynamical system
withα∈Aut(A)and letβ∈Aut(A)pointwise commute withα
(that is,β(α(a)) =α(β(a))fora∈A). Use the universality of the