144 4. Constructions(3) There is an action I (called the gauge action) of 1l on 0(1i) such
that
/z(a) =a and /z(Be) = zSe
for every z E 1l, a EA and e E 7i.Identity (2) follows from the fact that a - CJy:(a) = ee,e E OC(F(1i)IH),
where a E IH with a 2 0 and e = ;iJ2 E 7i^00 IH· The other facts descend
from the Toeplitz-Pimsner algebra.
Example 4.6.10 (The Toeplitz algebra and Cuntz algebras). We view .e;
as a C-correspondence over C. Then, T(.e;) is generated by isometries
T1, ... , Tn (we write Ti = T 0 i for simplicity) with orthogonal ranges. One
checks that e = 1 - I:r= 1 TiTt is the rank-one projection onto C c :F(.e;)
and the corresponding vector state w = Ep n is called the vacuum state. The
ideal IH coincides with(['. and On = O(.e;) = T(.e;)/OC(F(1i)) is generated
by isometries S1, ... , Sn such that I:r=l SiSi = 1. The C-algebra T(C) is
the classical Toeplitz algebra - the C* -algebra generated by the unilateral
shift on £^2 (N) - and On are the celebrated Cuntz algebras.
Example 4.6.11 (Bimodules from u.c.p. maps). Let cp be a u.c.p. map on
a unital C* -algebra A such that ker cp does not contain a nonzero ideal. The
algebraic tensor product A 8 A is naturally an A-A bimodule, with left and
right actions given by a· (x Q9 y) · b = ax Q9 yb. We equip A 8 A with an
A-valued semi-inner product
(a1 Q9 b1, a2 Q9 b2) = b!cp(a!a2)b2.
(It is routine to check that the product is positive semidefinite.) Let 1i~ be
the C* -correspondence obtained from A 8 A by separation and completion.
As usual, we denote by ;;@b the vector in 7i~ that is represented by a Q9 b E
A 8 A. Let e = f®l. Since AeA has dense linear span in 7i~, the Toeplitz-
Pimsner algebra T(1i~) is generated by A and an isometry T =Te which
satisfies the relation T* aT = cp( a) for every a E A.
Example 4.tB.12 (Crossed products by Z). Now suppose that cp is a *-
automorphism. Then, a -----Q9 b = 1 Q9 -cp(a)b for every a, b E A and ee,e = 1 in
IB(7i~). Hence, U = Se is a unitary element in the Cuntz-Pimsner algebra
0(1i~) which satisfies U*aU = cp(a) for a EA. It is not difficult to see that
0(1i~) ~A ><l!f' Z (especially after we prove universality).
Example 4.6.13 (Correspondences from graphs). Let <B = (V, E, s, r) be
a directed graph without a source^11 - i.e., there is no vertex v such that
(^11) To treat a graph with sources, one needs to work with correspondences whose left action
is not faithful.