Quantum Dynamical Systems from the Point of View of Noncommutative Mathematics 149
spectrum ofC). Using a straightforward ergodic type argument
using property (i) above, we see thatφmust be equal toτ|C. This
would have ended the proof if we knew that the CNT entropy is
monotone under passing to invariant subalgebras (see property
(ii) above), but this holds true only for so-called expected
subalgebras. Thus, we need to pass to the von Neumann algebra
set-up and exploit the fact thatτis tracial, implying that suitable
conditional expectations (that is, norm-one surjective projections
on the von Neumann subalgebras in question, preserving the trace
τ) always exist.
Note that the analogous question for the CNT entropy of the
automorphisms of the hyperfiniteI I 1 -factorRremains open – in
other words we do not know if there exists an automorphism ofR
whose CNT entropy (with respect to the canonical traceτonR) is
greater than the supremum of respective measure entropies over
the restrictions to all invariant commutative von Neumann
subalgebras ofR.
Theorem 2.3.7 implies also a necessity of looking for other
techniques of estimating the Voiculescu entropy from below.
Possible ways are related to investigating the classical entropy of
the induced dynamics of the state space ofA([Ke]) or exploiting
the connections to the index theory of von Neumann algebras
([Sk 2 ], see also [Hi]).
2.3.5 Automorphism that leaves no non-trivial abelian
subalgebras invariant
In view of the discussion above it might be natural to ask whether
for any noncommutative dynamical system(A,α) there exists a
non-trivial (that is, different from C (^1) A) commutative
C∗-subalgebraC⊂Aleft globally invariant byα. The following
example, inspired by a suggestion of Kummerer, shows that this is ̈
not the case.
Before we formulate the example we need to recall some facts
regarding the spectrum of unitary operators (for the details see the
books [He] and [Na]). Every unitary operatorUon a separable
Hilbert space is isomorphic to a countable direct sum of the
operators of the formMz∈L^2 (T,μ), whereμis a measure on the
circle, and(Mz(f))(t) =t f(t)for anyt∈T,f∈L^2 (T,μ). As each
such measureμdecomposes into a discrete part, a part which is