We have decided to collect all the preliminary results needed for Ext-theory and Kasparov theory into a single chapter, even though not all of the results will be needed immediately. We have done this since the three sections of this chapter are closely related and it is more efficient to do everything at once.

In Chapter VII, only parts of Sections 12 and 13 will be needed. Section 14 is not required (except for 15.13) until KK-theory (Sections 17ff.), which also requires all of Sections 12 and 13.

The reader who so desires may skip over this chapter, returning on an ad hoc basis as needed.

12. Multiplier Algebras

12.1. Introduction

Recall [Pedersen 1979, 3.12] that the multiplier algebra M(A) of A is the maximal C*-algebra containing A as an essential ideal. The strict topology on M(A) is the topology generated by the seminorms |||x|||a = ||ax|| + ||xa|| for a ∈ A.

The outer multiplier algebra Q(A) of A is the quotient M(A)/A. We will write Q for the Calkin algebra Q(𝕂) = 𝔹/𝕂.

EXAMPLES 12.1.1.

• (a) If A is unital, then M(A) = A. M(A) is always unital, so if A is nonunital M(A) ≠ A. (In fact in this case M(A) is generally much larger than A: for example, M(A) is never separable if A is nonunital [Pedersen 1979, 3.12.12].)

• (b) If A = C0(X), then M(A) = C(ßX), where ßX is the Stone-Cech compactification of X. The strict topology on bounded subsets of M(A) is the topology of uniform convergence on compact subsets of X.