Abstract:

Let A be a C*algebra. Since the bidual of A can be considered as a W*algebra, this enables us to prove the following duality theorems: (i) There exists a bijection between the normclosed 2sided ideals of A and the normclosed invariant order ideals of A. (ii) There exists a bijection between the normclosed left ideals of A and the normclosed order ideals of A. (iii) There exists an order inverting bijection between the normclosed 2sided ideals of A and the weak*closed invariant faces of S(A), where S(A) is the state space of A. The object of the thesis is to verify the above observations and to give Stormer's solution to J. Dixmier's problem: if N and M are normclosed 2sided ideals of A, then (N + M)+ = N + + M+, where N+ and M+ denote the positive parts of N and M respectively. 