4.5. Compact group actions and graph C*-algebras 135and r(e) is the vertex at which it ends). An operator algebraist likes to
think of the vertices as being (pairwise orthogonal) projections on a Hilbert
space and each directed edge corresponding to a partial isometry going from
the source projection to something underneath the range projection.^8 It is
not hard to show that such projections and partial isometries can always
be constructed (on a suitable direct sum of Hilbert spaces) and hence we
can appeal to universal nonsense to construct a largest C* -algebra generated
by such elements. More precisely, we have the following result (see [163,
Proposition 1.21]).
Theorem 4.5.3. Let ® = (V, E, s, r) be a row finite^9 graph. Then there
exists a C*-algebra C*(®) with the following properties:
(1) for each vertex v EV there is a projection Pv EC*(®) and the Pv 's
are pairwise orthogonal;
(2) for each edge e E E there is a partial isometry Se E C*(®) such
that s:se = Ps(e);
(3) for each v E V, if r-^1 (v) i-0, then Pv = l::eEr-l(v) Ses: - i.e., Pv
is the (finite) sum of the range projections of the partial isometries
coming from edges going into v;
(4) C*(®) = C*({Pv : v E V}, {se : e E E}) and is universal in the
sense that for any collection of projections { Pv : v E V} and partial
isometries {Se : e E E} satisfying the three conditions above, there
is a *-homomorphism
C*(®)---+ C*( {Pv : v EV}, {Se: e EE})
such that Pv 1-+ Pv, for all v E V, and Se 1-+ Se for all e E E.The C-algebra C ( ®) is called the graph C -algebra of the graph ®. It
follows easily from universality that C(®) admits a canonical gauge action
'JI' ---+ Aut(C(®)). That is, if z E C has modulus one, then the partial
isometries {zse : e E E} satisfy the same relations as {Se : e E E}; hence
there is a -isomorphism 'Yz: C(®) ---+ C(®) such that Pv 1-+ Pv and Se 1-+
zse. One checks that we have point-norm continuity, hence a group action
of 'JI' on C(®). According to Theorem 4.5.2, we would know that C(®)
is nuclear if we could show the fixed point subalgebra of the gauge action
to be nuclear. If you have studied Cuntz algebras already, then this should
come as no surprise. (Actually, Cuntz algebras are nice examples of graph
8some papers have the partial isometries go the other way, but we prefer to travel in the
direction of the edge.
9Row finite means that every vertex has at most finitely many edges coming into it -i.e.,
r-l(v) is a finite subset of E, for all v EV. It is possible that a vertex has infinitely many edges
going out of it, however.