A New Proof of Mack's Characterization of PCS-algebras

Abstract

Let $A$ be a $C^*$-algebra and $K_A$ its Pedersen's ideal. $A$ is called a PCS-algebra if the multiplier $\Gamma(K_A)$ of $K_A$ is the multiplier $M(A)$ of $A$. J. Mack [5] characterized PCS-algebras by weak compactness on the spectrum of $A$. We give a new simple proof of this Mack's result using the concept of semicontinuity and N. C. Phillips' description of $\Gamma(K_A)$.