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On The Center Of Quasi-Central Banach Algebras With bounded Approximate Identity

Published online by Cambridge University Press:  20 November 2018

Sin-Ei Takahasi
Sin-Ei Takahasi*
Affiliation:
Ibaraki University, Mito, Ibaraki
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Let A be a quasi-central complex Banach algebra with a bounded approximate identity and Prim A the structure space of A. In [15], we have shown that every central double centralizer T on A can be represented as a bounded continuous complex-valued function ΦT on Prim A such that Tx + P = ΦT(P)(x + P) for all x ∈ A and P ∈ Primal when the center Z(A) of A is completely regular. Here x + P for P ∈ Prim A denotes the canonical image of x in A/P. In particular, in the case of quasi-central C*-algebras, this result is equivalent to the Dixmier's representation theorem of central double centralizers on C*-algebras (see [3, Section 2] and [9, Theorem 5]).

In this paper, it is shown that if Z(A) is completely regular then the space Prim A is locally quasi-compact and for each element z of Z(A), ΦLz vanishes at infinity, where Lz for z ∊ Z(i) is the central double centralizer on A defined by Lz(x) = zx for all x ∊ A.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 1 , 01 February 1981 , pp. 68 - 90
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Archbold, R. J., Density theorems for the center of C*–algebra, J. London Math. Soc. (2) 10 (1975), 187–197.Google Scholar
2. Bonsall, F. F. and Duncan, J., Complete normed algebras, Erg. d. Math. Band 80 (Springer, Berlin-Heidelberg-New York, 1973).Google Scholar
3. Busby, R. C., On structure spaces and extensions of C*–algebras, J. Functional Analysis 1 (1967), 370–377.Google Scholar
4. Curtis, P. and Figa-Talamanca, , Factorization theorems for Banach algebras in function algebras (Scott, Foresman and Co., Chicago, 111, 1966), 169–185.Google Scholar
5. Dauns, J. and Hofmann, K. H., Representation of rings by continuous sections, Mem. Amer. Math. Soc. 82 (1968).Google Scholar
6. Davenport, J. W., Multipliers on a Banach algebra with a bounded approximate identity, Pacific J. Math. 63 (1976), 131–135.Google Scholar
7. Delaroche, C., Sur les centres des C*–algèbres, Bull. Sci. Math. 92 (1968), 111–128.Google Scholar
8. Dixmier, J., Les C*–algèbres et leurs représentations, Cahiers Scientifique 24 (Gauthier– Villars, Paris, 1964).Google Scholar
9. Dixmier, J., Ideal center of a C*–algebra, Duke Math. J. 35 (1968), 375–382.Google Scholar
10. Dixon, P. G., Semiprime Banach algèbres, J. London Math. Soc. (2) 6 (1973), 676–678.Google Scholar
11. Elliott, G. and Olesen, D., A simple proof of the Dauns-Hofmann theorem, Math. Scand. 34 (1974), 231–234.Google Scholar
12. Rickart, C. E., General theory of Banach algebras (Van Nostrand, N.J., 1960).Google Scholar
13. Takahasi, S., A remark on Urysohns lemma for spectrums of C*–algebras, Bull. Fac. Sci. Ibaraki Univ. Ser. A 4 (1972), 33–35.Google Scholar
14. Takahasi, S., A note on the homomorphic images of the center of C*–algebras, Bull. Fac. Sci. Ibaraki Univ. Ser. A 6 (1974), 29–32.Google Scholar
15. Takahasi, S., Dixmier's representation theorem of central double centralizers on Banach algebras, Trans. Amer. Math. Soc. (to appear).Google Scholar