166 Noncommutative Mathematics for Quantum Systems̃s(γ,x) =γ(x)and thusx=Φ(x). These two facts combined show
thatΦis an idempotent map, with the image equal to Fixα. The fact
thatΦis positive is a consequence of the fact that all the operations
involved in the construction are positivity-preserving (the reader is
invited to verify the details).
Finally for anyx∈Mwe haveω(Φ(x)) = ̃s(ω,x) =ω(x). This
ends the proof.
The theorem remains valid without the assumption on traciality of
ω: the general case can be reduced to the tracial one via the
Tomita–Takesaki theory (see for example Chapters VII–VIII of
[Ta 2 ]). The mapΦconstructed above can be naturally interpreted
as thenoncommutative conditional expectationonto Fixαpreserving
the stateω. It can be further shown to be completely positive and
normal (in fact each contractive idempotentEfrom aC∗-algebraA
onto aC∗-subalgebraBis automatically completely positive, and
is a module map with respect to a smaller subalgebra:
E(b 1 ab 2 ) =b 1 E(a)b 2 for alla∈A,b 1 ,b 2 ∈B– see Theorem 3.4 [Ta 1 ]- it only proves positivity but complete positivity can be shown
very similarly). It is also determined uniquely as a contractive
projection onto Fixαpreserving the stateω, which follows from
the next exercise.
Exercise 2.6.1 Use the information contained in brackets in the
last paragraph to prove that ifAis aC∗-algebra with a faithful
stateω ∈S(A)andBis aC∗-subalgebra ofAthen the norm-one
idempotentEfromAontoBpreservingω, if it exists, must be
unique. Would this be true if we do not assume thatEpreserves
ω? Can you find a (commutative, finite-dimensional)
counterexample?The last part of the above exercise partially explains the role of
the stateωin Theorem 2.6.1.2.6.2 Almost uniform convergence in von Neumann algebras
Extending the almost sure convergence of the individual ergodic
theorem to the noncommutative context is more problematic – we
first need to find a way of expressing it without using the ‘point’
picture.Theorem 2.6.2 (Egorov Theorem) A sequence (fn)∞n= 1 of
measurable functions on a probability space(Y,μ)is convergent