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Duality and the existence of weakly completely continuous elements in a B*-algebra

Published online by Cambridge University Press:  18 May 2009

B. J. Tomiuk
B. J. Tomiuk
Affiliation:
University of Ottawa, Ottawa, Canada
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Ogasawara and Yoshinaga [9] have shown that a B*-algebra is weakly completely continuous (w.c.c.) if and only if it is *-isomorphic to the B*(∞)-sum of algebras LC(HX), where each LC(HX)is the algebra of all compact linear operators on the Hilbert space Hx. As Kaplansky [5] has shown that a B*-algebra is B*-isomorphic to the B*(∞)-sum of algebras LC(HX) if and only if it is dual, it follows that a 5*-algebra A is w.c.c. if and only if it is dual. We have observed that, if only certain key elements of a B*-algebra A are w.c.c, then A is already dual. This observation constitutes our main theorem which goes as follows. A B*-algebra A is dual if and only if for every maximal modular left ideal M there exists a right identity modulo M that is w.c.c.

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 13 , Issue 1 , March 1972, pp. 56 - 60
Copyright
Copyright © Glasgow Mathematical Journal Trust 1972

References

REFERENCES

1.Bonsall, F. F. and Goldie, A. W., Annihilator algebras, Proc. London Math. Soc. (3) 4 (1954), 154–167.CrossRefGoogle Scholar
2.Bonsall, F. F. and Tomiuk, B. J., The semi-algebra generated by a compact linear operator, Proc. Edinburgh Math. Soc. (2) 14 (1965), 177–196.Google Scholar
3.Dixmier, J., C*-algebres et leurs représentations (Paris, 1964).Google Scholar
4.Dunford, N. and Schwartz, J. T., Linear operators, Part I (New York, 1958).Google Scholar
5.Kaplansky, I., Normed algebras, Duke Math. J. 16 (1949), 399–418.CrossRefGoogle Scholar
6.Kaplansky, I., The structure of certain operator algebras, Trans. Amer. Math. Soc. 70 (1951), 219–255.CrossRefGoogle Scholar
7.Köthe, G., Topologische lineare Räume (Springer-Verlag, 1960).CrossRefGoogle Scholar
8.Ogasawara, T., Finite dimensionality of certain Banach algebras, J. Sci. Hiroshima Univ. Ser. A 17 (1954), 359–364.Google Scholar
9.Ogasawara, T. and Yoshinaga, K., Weakly completely continuous Banach*-algebras, J. Sci.Hiroshima Univ. Ser. A 18 (1954), 15–36.Google Scholar
10.Rickart, C. E., General theory of Banach algebras (New York, 1960).Google Scholar
11.Sakai, S., Weakly compact operators on operator algebras, Pacific J. Math. 14 (1964), 659–664.CrossRefGoogle Scholar