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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.

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

References

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