154 Noncommutative Mathematics for Quantum Systems
crossed product (as alluded to above) to show that there exists a
unique automorphismβ ̃ofAoαZsuch that
(i)β ̃◦ι=ι◦β;
(ii)β ̃(u) =u.Observe that ̃α=Adu.
2.4.2 The Voiculescu entropy computations for the maps
extended to crossed products
One of the first natural questions regarding the Voiculescu entropy
of a given automorphism is whether it does not change under
passing to its natural ‘inner extension’ (that is, the extension to the
crossed product constructed as in Exercise 2.4.3). The following
result shows that this is indeed the case.
Theorem 2.4.2 ([Vo]) LetAbe a nuclearC∗-algebra, letα∈Aut(A)
and letβ ∈ Aut(A)pointwise commute withα. ThenAoαZis
nuclear and moreover htβ=ht ̃β.
Proof The fact thatAoαZis nuclear is a by-product of the proof
below; therefore, we will not focus on this part (although formally
we should first prove it so that we can formally speak of ht ̃β).
We first fixn∈Nand construct unital completely positive maps
ψn:AoαZ→Mn(A)andφn:Mn(A)→AoαZ. The first of them
is defined in the following way: letPn:^2 (Z)⊗H→^2 (Z)⊗H
be the orthogonal projection ontoLin{δk⊗ξ:k=1,... ,n,ξ∈H}.
Note that we can (and will) viewPnas a map from^2 (Z)⊗Hto Cn⊗H. AsAoαZ ⊂ B(^2 (Z)⊗H)we can consider the (unital,
completely positive) map
ψn(x) =PnxPn∗, x∈AoαZ.A prioriψntakes values inB(Cn⊗H)≈Mn(B(H)), but consider
the following computation(a∈A,ξ∈H,l∈Z,k∈{1,... ,n})
ψn(ι(a)ul)(δk⊗ξ) =Pnι(a)(δk−l⊗ξ) =Pn(δk−l⊗αl−k(a)(ξ))and the latter is equal either to 0 (ifk−l∈ {/ 1,... ,n}) or toδk−l⊗
αl−k(a)(ξ)(ifk−l∈ {1,... ,n}). Thus, using the picture of matrix
units as before we see that