ELEMENTARY OPERATORS AND NEW C -ALGEBRAS

Let H be a complex Hilbert space and B(H) the algebra of all bounded linear operators on H. In this paper, we study the class of pairs of operators A;B 2 B(H) that have the following property, ATB = T implies B TA = T for all T 2 C1(H) (trace class operators). The main result is the equivalence between this character and the fact that the ultra-weak closure of the range of the elementary operator A;B dened on B(H) by A;B(X) = AXB X is equivalent to the generalized quasiadjoint operators. Some new C -algebras generated by a pair of operators A;B2 B(H) are also presented.