# A NEW PROOF OF MACK'S CHARACTERIZATION OF PCS-ALGEBRAS

• Published : 2003.01.01
• 53 3

#### 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 $\Gamma(K_{A})$.

#### Keywords

PCS-algebra;multiplier;weakly compact set

#### References

1. $C^*$-algebras and their automorphism groups G. K. Pedersen
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3. Proc. Amer. Math. Soc. v.104 A new approach to the multipliers of Pedersen's ideal N. G. Phillips https://doi.org/10.2307/2046807
4. Duke Math. J. v.22 Applications of weak semicontinuity in $C^*$-algebra theory G. K. Pedersen
5. Proc. Amer. Math. Soc. v.9 On Properties characterizing pseudo-compact spaces R. W. Bagley;E. H. Connel;J. D. McKnight, Jr. https://doi.org/10.2307/2033015
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7. Canad. J. Math. v.40 Semicontinuity and multipliers of $C^*$-algebras L. G. Brown https://doi.org/10.4153/CJM-1988-038-5
8. The spectrum of a PCS-algebra J. Mack