1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

8.4. Homotopy invariance 275

Proposition 8.4.2. Let I be a nonunital C* -algebra. Then there exists a

unital C*-algebra M(I), called the multiplier algebra of I, with the property

that M (I) is the largest C* -algebra containing I as an essential ideal -i.e.,

if I <l A and I is essential, then there is a unique embedding A <-t M(I)

which extends the canonical inclusion I c M (I). Moreover, M (I) is the

unique (up to isomorphism) algebra with this property.

Proof. Let I c lE(7-i) be any nondegenerate representation and define

M(I) ={TE lE(7-i) : Tx EI and xT EI, for all x EI}.

Since I C lE(7-i) is nondegenerate, I sits in M(I) as an essential ideal (an

orthogonal element would give a subspace of 1-i where I restricts to zero).

Now suppose I <l A. Then the map I <-t lE(7-i) extends uniquely to a

*-homomorphism Jr: A~ lE(7-i) by defining

n(a) = limaen

where {en} C I is any approximate unit and the limit is taken in the strong
operator topology (uniqueness of Jr is again due to the fact that I C lE(7-i)
is nondegenerate). If I <l A is essential, then Jr must be injective -the kernel
of Jr will be orthogonal to I - and hence we must show that n(A) c M(I).
However, if we take {en} c I to be quasicentral, then this is easily verified.
Indeed,

n(a)x = lim(aen)x = lima(enx) =ax EI

and multiplication on the other side is similar since {en} is quasicentral.

Uniqueness of M(I) follows easily from the universal property. D

Lemma 8.4.3. Suppose I c J is an inclusion of nonunital C* -algebras such
that some approximate unit {en} CI is also an approximate unit of J. Then
we have a natural inclusion M (I) c M ( J) such that M (I) n J = I.

Proof. Let Jc lE(7-i) be nondegenerate. Then I c lE(7-i) is also nondegen-
erate and hence I c M(I) C lE(7-i). Let TE M(I) and x E J be arbitrary.
Then

Tx = limT(enx) = lim(Ten)x,

since {en} C I is an approximate unit of J. But Ten E I and thus (Ten)x E
J for all n - i.e., T E M(J). Now assume that x E M(I) n J. Then
llx - enxll ~ 0 implies x E I, since enx E I by assumption. D

Lemma 8.4.4. Assume I is AF embeddable. Then there is an inclusion
I c J where J is AF and some approximate unit {en} C I is also an
approximate unit of J. (Hence M(I) C M(J) and M(I) n J =I.)