Quantum Dynamical Systems from the Point of View of Noncommutative Mathematics 131
(i) eachλγis unitary;(ii) for anyγ,γ′∈Γwe haveλγλγ′=λγγ′.Definition 2.1.12 The C∗-algebra generated inB(`^2 (Γ))by the
collection of operators{λγ}γ∈Γis called the reducedC∗-algebra of
Γand denoted byCr∗(Γ).
Note that the Cuntz algebras and graphC∗-algebras introduced
above were defined asabstract C∗-algebras, that is, completions of
some ∗-algebras with respect to the universal C∗-norm. The
algebraC∗r(Γ) is defined as a concrete C∗-algebra, that is, as a
specific closed unital∗-subalgebra ofB(H)for a specific Hilbert
spaceH. It is possible to associate withΓalso a natural abstract
C∗-algebra, the universal (or full) C∗-algebra of Γ. It is denoted
simply asC∗(Γ)and arises as the universalC∗-completion of the
purely algebraic group ringC[Γ]. AsCr∗(Γ)can be viewed asa
C∗-completion ofC[Γ], the definitions imply the existence of the
canonical surjective unital∗-homomorphismπ:C∗(Γ)→C∗r(Γ).
Note, however, thatπneed not be injective (it is injective if and
only ifΓis amenable, see Section 2.4).
WhenΓis abelian, thenCr∗(Γ)is commutative and isomorphic
to theC∗-algebraC(Γˆ), whereΓˆis a compact space, thedual group
(thePontriagin dual) ofΓ. This is essentially a Fourier transform
fact; note that it can be viewed as a starting point to introduce the
noncommutative objects ‘dual’ to classical, not necessarily abelian,
discrete groups.
Exercise 2.1.7 Decribe explicitly the isomorphism betweenCr∗(Γ)
andC(Γˆ)whose existence is stated above for finite cyclic groups
(Z/nZ,n∈N). Remember thatẐ/nZ=Z/nZ.
2.2 Noncommutative Topological Entropy of Voiculescu
In this section, we introduce certain classes of quantum dynamical
systems illustrating the general theory, recall the classical notion
of topological entropy, and introduce its noncommutative counter
part, Voiculescu entropy.