4.6. Ountz-Pimsner algebras 141Theorem 4.6.4. Let A be a C*-algebra and X be a Banach A-module.
Then, the subset
AX = {ax : a E A, x E X} c X
is a closed A-submodule of X.Proof. (Pedersen) Letting Y be the closed linear span of AX, we'll show
Y = AX. It is easily verified that for every y E Y and approximate unit
(ei) of A, we have eiy--+ y. This implies Y can be viewed as a module over
the unitization A. Indeed, for every a E A, ,\ E <C and y E Y, we have
[[(a+ .\l)y[[ = lim l[(a + .\l)eiyl[::; lim [[(a+ .\l)eil[llYll =Ila+ .\lllllYll·
Let z E Y and E > 0 be given. Set ao = lA, Yo= z and define inductively
an= an-1 - 2-n(l - en) EA, Yn = a;;-^1 z E Y.Then, an= 2-n + 2=~= 1 2-kek 2: 2-n and hence 1[2-na;-^111 ::; 1. Moreover,
Yn - Yn-1 = a;;-^1 (an-1 - an)Yn-1 = 2-na;;-^1 (1-en)Yn-1·
Choosing the en's carefully, we can assume that llYn - Yn-1 II < 2-nE for all
n. Let a = liman = z=~ 1 2-kek E A and y = limyn E Y. (Note that
0::; a::; 1, llY - z[[ < E.) Then z = limanYn = ay, as desired. D
Let 1i be a C* -correspondence over B and J <l B be an ideal. We set
1-iJ = {(b: ( E 7-i, b E J}.By Cohen's factorization theorem, 1-iJ is a closed right B-submodule of 7-i.
Hence, 1-iJ is again a C*-correspondence over B. Moreover, ( E 1-iJ if and
only if ( (, () E J (one direction is trivial and the other uses approximate
units).
Toeplitz and Cuntz-Pimsner algebras. Now we move to the main topic
of this section. Let 1i be a C -correspondence over A. For notational con-
venience, we view A as a subalgebra of IB(7-i) and stop writing 7r?-l· We set
1-i®O =A and 1-i®n = 1i @A··· @A 1-i, then-fold tensor product. The full
Fack space over 1i is the C -correspondence :F('Ji) over A, defined by
:F('Ji) = EB 7-i®n.
Abusing notation, we also view A C JIB(:F(7-i)) omitting 1fJ=(1i)· Via the
identification 1i @A :F('Ji) ~ ffin> 1 1i®n C :F('Ji), for each ~ E 1i we define
Tf, E JIB(:F(7-i)): -
Tf,(a) =~a and Tf,(6 Q9 • • ·@ ~n) = ~@ 6@ · · ·@ ~n·
These operators Tf, are called creation operators and satisfy the relations
Taf.+7J = 01.Tf, + T'T/, Taf.b = aTf,b, and TgT'T/ = (~, 17)