142 4. Constructions
for every a E ('., e,'f] E 1i and a,b EA. Let Oj:: JIB(H)-+ JIB(F(H)) be the
*-representation given by cr;:-(x) = 01-l®o EB (x 01;:-(?-l)); i.e.,
cr;:(x)a = 0 and cr;:-(x)(6 0 ... 0 en)= (x6) 0 6 0 ... 0 en·
Note that cr;:-JK(?-l) is nothing but the representation CTT defined in Propo-
sition 4.6.3. Let Po E JIB(F(H)) be the orthogonal projection onto 1i®^0.
Evidently Po commutes with every a E A and
a= PoaPo +er;:( a) E JIB(F(H)).
For a E H'g,o, we set Ta,= a E JIB(F(H)); forμ= 6 0 ···@em E H®m, we set
Tμ = Tfa · · · Tt:,rn E JIB(F(H)). Observe that this notation is compatible with
Tμ: F(H)-+ H®m 0A F(H) ~ EB H®n C F(H).
n?_mThen, for every μ E H®m and v E H®n ( m, n 2:: 0), we have
Bμ,v = TμPoT: E JK(F(H)).
Definition 4.6.5. Let 1i be a C-correspondence over A. The (augmented)
Toeplitz-Pimsner algebra T(H) is the C-subalgebra of JIB(F(H)) generated
by A and {Tt:, : e E H}.^10
We record the main identities that hold in T(H).Theorem 4.6.6. LetT(H) = C(AU{Tt:,: e EH}) be the Toeplitz-Pimsner
algebra of a C -correspondence 1i over A.
(1) For every a EC, e, 'fl E 1i and a, b EA, we have
Tott:,+'f} = aT1:, + T'f/, Tat:,b = aT1:,b and T!T'f/ = (e, rJ).
(2) We have T(H) = span{TμT: : μ E H®m, v E H®n, m, n 2:: O} and
there exists a nondegenerate conditional expectation E1-l from T(H)
onto A such that E?-l(TμT:) = 0 for everyμ E H®m and v E H®n
with (m, n) =/= 0.
(3) There is an action/ (called the gauge action) of 11' on T(H) such
that
rz(a) =a and /z(Tt:,) = zT1:,
for every z E 11' = {z E ('.: JzJ = 1}, a EA and e E 1{.
Proof. The conditional expectation E1-l: T(H) -+A in assertion (2) is given
by compression to H®^0 :
E1-l(x) = PoxPo E JIB(H®^0 ) ~Ac T(H).
(^10) This definition is not exactly the same as Pimsner's in the nonfull case -i.e., when { (/:,, 'TJ) :
I:,, 'f} E 7-l} does not generate A -hence the (augmented) terminology. However, most examples
are full, and then our definition agrees with Pimsner's.