- Cuntz-Pimsner algebras 145
r-^1 (v) = 0. Then the set coo(E) of all finitely supported functions on Eis
naturally a co(V)-bimodule, with the left and right action given by
(f. e. g)(e) = f(r(e))e(e)g(s(e))
fore E coo(E) and f, g E co(V). We equip Coo(E) with a co(V)-valued inner
product by
\e,77)(v) = I:: e(e)77(e).
eEs-^1 (v)
(If v E Vis a sink, i.e., s-^1 (v) = 0, we set \e,77)(v) = 0.) It is routine
to check that coo(E) gives rise to a C-correspondence H® over c 0 (V) by
separation and completion. The C -correspondence H® is full if and only if
the graph does not have a sink. We now assume that the graph is row finite.
Then,
8v = L Boe,Oe E K(H®)
eEr-^1 (v)
for every v E V. The Cuntz-Pimsner algebra O(H®) is generated by pro-
jections { 8v : v E V} and partial isometries {Boe : e E E} such that
B"teBoe = 8s(e) and L BoeB"te = 8v.
eEr-^1 (v)
Therefore, by universality, there exists a -homomorphism from the graph
C-algebra C(<B) onto O(H®) which maps the Pv's to the 8v's and the se's
to the Boe 's. It follows from gauge-invariant uniqueness (Theorem 4.6.20
below) that this -homomorphism is actually a *-isomorphism.
Universality of Pimsner algebras.
Definition 4.6.14. Let H be a C-correspondence over A. A represen-
tation of Hon a C-algebra Bis a pair (7r,r), where 7r: A, Bis a*-
homomorphism and r: H , B is a linear map such that
r(aeb) = 7r(a)r(e)7r(b) and r(e)*r(77) = 7r( \e, 77) ).We denote by C(7r, r) the C-subalgebra of B generated by 7r(A) and r(H).
A representation (if, i) of H is universal if for any other representation ( 1f, r)
of H, there is a (continuous) -homomorphism from C(if,i) to C*(7r, r)
sending if(a) to 7r(a) and i(e) to r(e).
We observe that if (7r, r) is a representation of H, then
llr(e) II = 111f( \e, e)) 11112 ::::; 11e11
for every e E H. In particular, r is isometric if 1f is. Thus, considering a
suitable direct sum, one can show that a universal representation (if, i) of
H always exists; we will see that the canonical representation ( 1f :F(1i), T) of
Hon T(H) is universal.