152 Noncommutative Mathematics for Quantum Systems
2.4 Crossed Products and the Entropy
In this section we introduce the crossed product construction,
which in its simplest form we will encounter here may be viewed
as means of ‘making’ a given automorphism of aC∗-algebra inner
- albeit in a largerC∗-algebra. Then we discuss a natural extension
procedure for a map on aC∗-algebra commuting with the initial
automorphism to the crossed product and show that this
procedure does not change the Voiculescu entropy.
2.4.1 Crossed products
Let (α,A) be a noncommutative dynamical system, with
α∈Aut(A). Recall that an automorphismαis calledinnerif there
exists a unitaryu∈Asuch that
α(a) =u∗au, u∈A.We then writeα=Adu. Note that usually not all automorphisms
of a givenAare inner (consider the example ofA = C(X)for a
compact spaceX).
Below we describe a certain natural construction (going back to
Murray and von Neumann), which, given a pair(α,A), yields a
noncommutative dynamical system(β,B)‘extending’(α,A)and
such thatβ ∈ Aut(B)is inner. Assume thatAis faithfully and
non-degenerately represented on a Hilbert spaceHand consider
the Hilbert space^2 (Z)⊗H, which we can identify with the space of all sequences (ξn)n∈Z of vectors in H for which ∑n∈Z ‖ξn‖^2 <∞. For eacha∈Aconsider the following operatorι(a)on ^2 (Z)⊗H:
ι(a)((ξn)n∈Z) = (α−n(a)ξn)n∈Z, (ξn)n∈Z∈`^2 (Z)⊗H.It is easy to check that eachι(a)is bounded and further that the map
ι:A→B(^2 (Z)⊗H)is a unital injective representation ofA. Let furtheru∈B(^2 (Z)⊗H)denote the amplification of the two-sided
shift map on`^2 (Z):
u((ξn)n∈Z) = (ξn+ 1 )n∈Z, (ξn)n∈Z∈`^2 (Z)⊗H;The operatoruis of course a unitary. TheC∗-algebra generated by
π(A)anduinB(`^2 (Z)⊗H)is called thecrossed product ofAbyα
and is usually denotedAoαZ. The notation reflects the fact that