128 Noncommutative Mathematics for Quantum Systems
suppose thatXis a closed subset of a compact spaceY. What can
be then said of algebrasC(X)andC(Y)?
Exercise 2.1.4 Show that Lin{SμS∗ν:μ∈ Jk,ν∈ Jk,k∈N}is the
union of an increasing sequence of finite-dimensional unital
∗-algebrasFl
N = Lin{SμS
∗
ν : μ ∈ Jk,ν ∈ Jk,k ≤ l}(l ∈N).
IdentifyFlNwith a familiar (by now) C∗-algebra and interpret
Lin{SμS∗μ:μ∈Jk,k≤l}as a subalgebra ofFNl.
2.1.4 GraphC∗-algebras
Cuntz algebras turn out to be special examples of a very important
and wide class of C∗-algebras associated to (finite) directed
graphs. They first appeared in literature as generalizations of
Cuntz–Krieger algebras (to be described in what follows), which
in turn generalize Cuntz algebras introduced above.
LetΛ= (V,E,r,s)be afinite directed graph. By that we mean that
VandEare finite sets (thought of, respectively, as sets of vertices
and edges ofΛ) andr,s:E→Vcertain functions (thought of as
attributing to each edge its source and range vertex). A (finite)path
inΛis a finite sequence of edges, sayν:= (ei)ki= 1 such that for all
i=1,···,k−1 we haves(ei) =r(ei+ 1 ). The numberkis called
the length of the pathν(which we write as|μ|=k); the pathνhas
therange r(ν):=r(e 1 )and thesource s(ν):=s(ek). Thus, edges can
be viewed as paths of length 1; moreover, we can view vertices as
paths of length 0. Compatible paths can be concatenated: ifμandν
are paths, ands(μ) =r(ν), then the pathμνis given by consecutive
enumeration of edges belonging toμfollowed by the enumeration
of edges belonging toν. The set of all finite paths (including paths
of length 0) will be denoted byPΛ. We say thatΛcontains no sources
if for eachv∈Vand eachm∈Nthere exists a pathμof length
msuch thatr(μ) = v. Generalizing the idea of paths leads to the
notion of the space of infinite paths,PΛ∞: these are functions from
N 0 toEsuch that for alli∈N 0 the source of thei-th edge is equal
to the range of thei+1-th edge. Note that if the set of edges has
cardinalityN, thenPΛ∞is a (closed) subset of the Cantor setCN;
moreover,PΛ∞is non-empty ifΛcontains no sources.
A collection of operators{Pv:v∈V}∪{Se:e∈E}in some
C∗-algebra is called a non-degenerate Cuntz–Krieger family
(associated withΛ) if the following conditions hold (v∈V,e∈E):