Noncommutative Mathematics for Quantum Systems

Quantum Dynamical Systems from the Point of View of Noncommutative Mathematics 151

We will now use the assumption thatαleavesCinvariant. This
implies that for eachi∈Nthere existski ∈Nsuch thatα(Qi) =
Qki. Indeed,α(Qi)must be a projection inC(of the same rank as

Qi) and consideringα−^1 we see thatα(Qi)is also minimal (inC).
The mapN 3 i7→kiis a permutation ofN, which we will denote
byσ.
Recall now the concrete form ofα. The equalityα(Qi) = Qki
means that the unitaryU∗(whose spectral type is the same as that
ofU) maps the spaceQiHontoQσ(i)H. We then need to consider
two cases. First suppose thatσcontains a finite cycle, sayl 1 ,... ,lk.

Then the spaceK:=

⊕k
i= 1 QliHis a finite-dimensional subspace of
Hinvariant forUand the spectral theorem for matrices implies
that U must have an eigenvector inside K. This is however
impossible, as the point spectrum ofU∗is empty.
It remains then to consider the case whenσcontains an infinite
orbit. Chooseklying in such an orbit and a norm-one vector inQkH,
sayξ. Putξk=Ukξfor allξ∈Z. Then the spaceK′:=Lin{ξk:k∈
Z}is left invariant byU, and asUξk=ξk+ 1 (k∈Z), the restriction
ofUtoK′is unitarily equivalent to the usual two-sided shiftSon
`^2 (Z). This contradicts the remarks before the theorem.

It may seem that the last theorem is likely to provide an alternative
to Theorem 2.3.7 in offering an example of an automorphism of a
nuclearC∗-algebra whose Voiculescu entropy is strictly greater
than the supremum of classical entropies of all of its classical
subsystems. This is however not true, as any automorphismαof
KH+C1 has Voiculescu entropy equal to 0. This is proved in
Example 6.2.10 (i) of [NS] – the argument is based on the fact that
it suffices to consider the setsΩsupported on finite-dimensional
subspaces ofH, saypH, wherepis a finite rank projection. Then
forn∈Nthe set

⋃n− 1

k= 0 α

k(Ω)is supported in the spacepnH, where

pn=

∨n− 1

k= 0 α

k(p)and one can build an (exact) approximation via

pnB(H)pn(strictly speaking we also need to cater forC1, but that
is easy). The latter algebra has rank equal to the rank ofpn, which
cannot be greater than n times rank of p. Thus, the size of
approximations grows at most linearly and htα=0.