130 Noncommutative Mathematics for Quantum Systems
As in the case of Cuntz algebras, we will be interested in what
follows in some subalgebras ofC∗(Λ). In particular, define
FΛ=Lin{SμS∗ν:μ,ν∈PΛ,|μ|=|ν|},CΛ=Lin{SμS∗μ:μ∈PΛ}.The second of the algebras above can be shown to be
commutative; in fact, following arguments similar to those of
Theorem 2.1.10 one can prove that it is isomorphic to the algebra
of continuous functions onPΛ∞. We will denote the corresponding
isomorphism fromCΛ toC(PΛ∞) byγΛ. The structure ofCΛis
slightly more complicated to analyze — it is however always an
AF-algebra, that is, an inductive limit of a sequence of finite direct
sums of matrix algebras.
The construction described in this section has far-reaching
extensions to the case of infinite graphs, tohigher-rank graphsof
Kumjian and Pask, which can be viewed as colored graphs
equipped with special rules identifying certain paths built of
edges of different colors, or to topological graphs. A rich and
accessible source of information about various graphC∗-algebras
can be found in the monograph [Ra].
2.1.5C∗-algebras associated with discrete groups
In the following sections we will use a few times examples based
on certainC∗-algebras associated with discrete groups. Here we
introduce the relevant construction.
LetΓbe a discrete group and consider the Hilbert space^2 (Γ). Each elementγ∈Γdefines aleft shifton^2 (Γ), which is a bounded
operatorλγ∈B(`^2 (Γ)), by the formula
(
(λγ)(f))
(γ′) =f(γ−^1 γ′), f∈`^2 (Γ),γ′∈Γ.It is easier to understand the definition ofλγif we write how it acts
on the elements of the typeδγ′∈`^2 (Γ), which form an orthonormal
basis in`^2 (Γ):
λγδγ′=δγγ′, γ,γ′∈Γ. (2.1.2)Then it is easy to see that the collection of operators (λγ)γ∈Γ
satisfies the following properties: