1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

140 4. Constructions

where T~ = [T6 · · · T~nl E JIB(AEEin, 1-i) and likewise for Try. Note that for
operators x and y,
llxyll = llyxxylll/2 = ll(xx)1/2yy(xx)1/2ll1/2 = ll(xx)1/2(yy)1/211.
Thus,
n
II L e~i.?JJ = ll(TfTt)^112 (r;rij)^11211 = ll[(ei, ej)lif
2
[(1Ji, 1Jj)lif
2
11.
i=l
This completes the proof. D
Remark 4.6.~:. Note the following special case: 11e~.~ II = 11e11^2 ' for all e E _H.
It follows that if (ei) is an approximate unit for m:::(H), then
lie - eiell^2 = llOg-eig,~-eigll = llOg,g - e~,gei - eieg,g + eieg,geill ---+ 0,
for every e E 1-l.
Proposition 4.6.3. Let 1-l be a Hilbert A-module and B be a C -algebra.
Let 7r: A ---+ B be a -homomorphism and r: 1{ ---+ B be a linear map such
that r(e)r(17) = 7r( (~, 17)) for every~' 'T/ E 1-l. Then, the -homomorphism
(}" 7 : OC(N)---+ B, defined by
n n
L:egi,?Ji H l:r(~i)r(1Ji)*,
i=l i=l
is continuous and satisfies (}" 7 (x)r(~) = r(x~) for every x E OC(H) and~ E 1-l.
Moreover, (}" 7 is injective whenever 7r is injective.

Proof. It is not hard to see that r(~a) = r(~)7r(a) for every~ E 1{ and a EA

and that (}" 7 is a well-defined *-homomorphism on the *-subalgebra of "finite-

rank operators." Let 6, ... , en, 'T/1, ... , 'T/n E 1i be given. By Lemma 4.6.1,

we have n

II I: e~i.11J = ll[(ei, ~j)Jij

2

[(rJi, 'T/j)Jtj

2

11.

i=l

On the other hand, the proof of Lemma 4.6.1 shows

n

II L r(~i)r(rJi)*ll =II [r(~i)*r(~j)Jif

2

[r(rJi)*r(rJj)]ij

2

ll

i=l

n

= 117r([(ei, ~j)lif

2

[(1Ji, 1Jj)lif

2
) II ~II I: egi,11J·
i=l
This shows that (}" 7 is contractive and that (}" 7 is isometric whenever 7r is. D

Let A be a C*-algebra. A (left) Banach A-module is a Banach space
X which is an A-module satisfying llaxll ~ llall llxll for every a E A and
x E X. The following is a special case of the Cohen factorization theorem
for Banach modules.