162 Noncommutative Mathematics for Quantum Systems
is a (normal) isomorphism of the von Neumann algebraMand
πω(M)′′=πω(M). Moreover, the collection of functionals{M 3 x
→〈ξ,πω(x)η〉:ξ,η∈Hω}spansM∗.
We finish this section by returning once again to the example of
the group von Neumann algebra for a discrete groupΓ. Consider
the vector functional on VN(Γ)given by the formula
τ(x) =〈δe,xδe〉, x∈VN(Γ), (2.5.2)whereedenotes the neutral element ofΓ.
Proposition 2.5.10 The vector functionalτdefined in (2.5.2) is a
faithful normal trace on VN(Γ), called thecanonical traceon VN(Γ).
Proof The fact thatτis a state is easy to verify, asτ(x∗x) =‖xδe‖^2
for anyx∈VN(Γ). Its normality is a consequence of one of the
facts stated before Proposition 2.5.8. To show faithfulness we need
first to note that elements of VN(Γ)commute with the right shifts
{rγ:γ∈Γ}⊂B(`^2 (Γ)), where
rγ(δγ′) =δγ′γ− 1 , γ,γ′∈Γ(compare the above formula with formula (2.1.2)). As VN(Γ) =
C∗r(Γ)′′, it suffices to show that elements ofCr∗(Γ)commute with
the right shifts; by linearity and continuity it further suffices to
show that left shifts commute with right ones, that is,
λγrγ′ =rγ′λγfor allγ,γ′ ∈ Γ. The last formula can be verified
directly.
Suppose then thatx∈VN(Γ)andτ(x∗x) =0. This means that
‖xδe‖^2 = 0, soxδe = 0. However, then for anyγ ∈Γwe have
0 =rγ− 1 xδe=xrγ− 1 δe=xδγ, so thatx=0.
To verify the tracial property ofτit again suffices because of
linearity and continuity to check that for anyγ,γ′ ∈Γwe have
τ(λγλγ′) = τ(λγ′λγ). This however is equivalent with the
statement
γγ′=e⇐⇒γ′γ=e,which is clearly true.
Exercise 2.5.3 Use the notation of the above proposition and
denote the von Neumann algebra generated inB(`^2 (Γ))by the