150 Noncommutative Mathematics for Quantum Systems
absolutely continuous with respect to the Lebesgue measure and
the remainder (the singular part), we obtain the analogous
decomposition ofU; moreover, this decomposition is unique in a
natural sense. The discrete part corresponds to thediscrete spectrum
ofU (that is, the collection of eigenvalues of U). A standard
example of a unitary with the Lebesgue continuous spectrum is
the two-sided shift on`^2 (Z), which is isomorphic toMzacting on
L^2 (T,λ)withλthe Lebesgue measure. Finally, we say thatUhas
purely singular spectrumif all the measures μ appearing in the
decomposition are singular. The uniqueness mentioned above
implies that such aUadmits no eigenvalues and no invariant
subspace on which it has Lebesgue continuous spectrum
(examples can be constructed of course by looking at Mz on
L^2 (T,μ)for a fixed singular measure onT).
Theorem 2.3.8 LetHbe a separable infinite-dimensional Hilbert
space andU∈B(H)be a unitary with purely singular continuous
spectrum. Consider theC∗-algebraA=KH+C 1 ⊂B(H)and the
automorphism ofAgiven byα(x) =U∗xU,x∈A. ThenC1 is the
only commutative non-trivialC∗-subalgebra left invariant byα.
Proof Let U be as above. Suppose that C is a commutative
C∗-subal–gebra ofAleft invariant byα. ThenC∩KHis also left
invariant byα; therefore, we may assume thatC⊂ KH(thus, we
need to deduce thatC={ 0 }).
We begin by understanding how commutative subalgebrasCof
KHlook like. Any operatorx∈Cis normal. As it is also compact, it
is of the form∑ki= 1 λiPi, wherePiare finite-rank projections andλi
are non-zero complex numbers, and eitherk=∞, in which case the
sequence(λi)∞i= 0 converges to 0, ork∈N(we can assume thatx 6 =
0). Functional calculus implies that eachPibelongs toC. This means
thatCis spanned (in the sense described above) by its projections.
In the next step we consider the family
PminC :={p∈C:pminimal non-zero projection inC}.Elements ofPCminare mutually orthogonal (can you see why?) and
can be listed in the sequence (Qi)∞i= 1 (remember that H is
separable)^1. It follows thatC={∑i∞= 1 λiQi:(λi)∞i= 0 ∈c 0 }.
(^1) Strictly speaking the sequence could be finite, but then the following arguments would
also work – it would suffice to consider the case of a finite cycle.