Geodesic
Duality and the existence of weakly completely continuous elements in a B*-algebra
Glasgow mathematical journal, Tome 13 (1972) no. 1, pp. 56-60

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Tomiuk, B. J. Duality and the existence of weakly completely continuous elements in a B*-algebra. Glasgow mathematical journal, Tome 13 (1972) no. 1, pp. 56-60. doi: 10.1017/S0017089500001373
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