Order ideals in a C*-algebra.

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 norm-closed 2-sided ideals of A and the norm-closed invariant order ideals of A. (ii) There exists a bijection between the norm-closed left ideals of A and the norm-closed order ideals of A. (iii) There exists an order inverting bijection between the norm-closed 2-sided 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 norm-closed 2-sided ideals of A, then (N + M)+ = N + + M+, where N+ and M+ denote the positive parts of N and M respectively.