Some characterizations of duality in B*-algebras.

Abstract

The purpose of the thesis is to assemble together the various known charracterizations of duality in B*-algebras. Dual B*-algebras have first been studied by I. Kaplansky. He obtained several characterizations of duality in B*-algebras. He showed for example that a B*-algebra is dual if and only if it is a closed *-subalgebra of the algebra LC(H) of all compact linear operators on a complex Hilbert space H. Also that a B*-algebra A is dual if and only if the socle of A is dense in A. The rest of the thesis is concerned mainly with results obtained by B. J. Tomiuk, T. Ogasawara and K. Yoshinaga on dual B*-algebras. We show that a B*-algebra is dual if and only if it is complemented or w.c.c.. We also show that a B*-algebra A is dual if and only if every maximal commutative *-subalgebra of A is dual. We end the thesis with a discussion about the successive conjugate spaces of a dual B*-algebra.