Quantum Dynamical Systems from the Point of View of Noncommutative Mathematics 129
(i) eachPvis an orthogonal projection, that is,Pv=Pv∗=Pv^2 ;(ii)Se∗Se=Ps(e)(in particular eachSeis apartial isometry, that is
SeSe∗Se=Se);(iii)Pv=∑f:r(f)=vSfSf∗;(iv)∑w∈VPw=1.Note that the relations above imply, for example, that ife,f ∈ E
ande 6 = f, thenSe∗Sf =0 (consider separately the cases when
r(e) =r(f)andr(e) 6 =r(f)). Similarly, one can show thatSeSf= 0
ife,f∈E,r(f) 6 =s(e). This suggests that we can usefully define for
a given Cuntz–Krieger family also partial isometries corresponding
to paths inPΛ: ifν:= (ei)ki= 1 is a path, we putSν=Se 1 ···Sek.
Definition 2.1.11 The graph C∗-algebraC∗(Λ) associated to a
finite directed graphΛwith no sources is the universalC∗-algebra
generated by a non-degenerate Cuntz–Krieger family.
The above definition has to be understood precisely in the way
in which the Cuntz algebra was introduced in Definition 2.1.8. In
particular,C∗(Λ) is equipped with a canonical non-degenerate
Cuntz–Krieger family{Pv:v∈V}∪{Se:e∈E}; moreover, using
the commutation relations satisfied by Cuntz–Krieger families one
can show that the set{SνS∗μ : μ,ν ∈ PΛ}is linearly dense in
C∗(Λ).
Exercise 2.1.5 Show that ifN≥2 andΛis a graph with a single
vertex andNedges, thenC∗(Λ) ≈ ON. Analyse the connections
between paths and multi-indices.
Exercise 2.1.6 Experiment with some graphs having two vertices
that contain no sources and see if you can identify the associated
graphC∗-algebras.
Another important class of graphC∗-algebras is determined by
{0, 1}-valued matrices: ifk ∈ NandM ∈ Mk({0, 1})then the
asssociated graphΛMis given by the set ofkvertices and the
collection of edges determined by the condition(i,j) ∈Eif and
only ifM(i,j) =1. The resulting graphC∗-algebra is called the
Cuntz–Krieger algebraassociated toM.